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Graphical degree sequences and Definitions realizations and - - PowerPoint PPT Presentation

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Swap- distances P .L. Erds Graphical degree sequences and Definitions realizations and History Undirected swap- sequences Pter L. Erds Bipartite degree sequences Alfrd Rnyi Institute of Mathematics Directed Hungarian

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Graphical degree sequences and realizations

Péter L. Erdös

Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences

MAPCON’12 MPIPKS - Dresden, May 15, 2012

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Graphical degree sequences and realizations

Péter L. Erdös

Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Joint work with Zoltán Király and István Miklós

MAPCON’12 MPIPKS - Dresden, May 15, 2012

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

1

Definitions and History

2

Undirected swap-sequences

3

Bipartite degree sequences

4

Directed degree sequences

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences

G(V; E) simple graph; V = {v1, v2, . . . , vn} nodes positive integers d = (d1, d2, . . . , dn).

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SLIDE 5

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences

G(V; E) simple graph; V = {v1, v2, . . . , vn} nodes positive integers d = (d1, d2, . . . , dn). If ∃ simple graph G(V, E) with d(G) = d

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SLIDE 6

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences

G(V; E) simple graph; V = {v1, v2, . . . , vn} nodes positive integers d = (d1, d2, . . . , dn). If ∃ simple graph G(V, E) with d(G) = d ⇒ d is a graphical sequence G realizes d.

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SLIDE 7

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences

G(V; E) simple graph; V = {v1, v2, . . . , vn} nodes positive integers d = (d1, d2, . . . , dn). If ∃ simple graph G(V, E) with d(G) = d ⇒ d is a graphical sequence G realizes d. Question: how to decide whether d is graphical?

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SLIDE 8

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences

G(V; E) simple graph; V = {v1, v2, . . . , vn} nodes positive integers d = (d1, d2, . . . , dn). If ∃ simple graph G(V, E) with d(G) = d ⇒ d is a graphical sequence G realizes d. Question: how to decide whether d is graphical?

  • Tutte’s f-factor theorem (1952) - applied for Kn
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences

G(V; E) simple graph; V = {v1, v2, . . . , vn} nodes positive integers d = (d1, d2, . . . , dn). If ∃ simple graph G(V, E) with d(G) = d ⇒ d is a graphical sequence G realizes d. Question: how to decide whether d is graphical?

  • Tutte’s f-factor theorem (1952) - applied for Kn

polynomial algorithm to decide

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

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SLIDE 11

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.
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SLIDE 12

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di)

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SLIDE 14

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps

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SLIDE 15

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps blacks ∈ G, reds (or blues) are missing

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps blacks ∈ G, reds (or blues) are missing ⇒ blacks are missing reds ∈ G′

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SLIDE 17

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps blacks ∈ G, reds (or blues) are missing ⇒ blacks are missing reds ∈ G′ The new realization satisfies the same degree sequence

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SLIDE 18

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps Let the lexicographic order on [n] × [n]

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SLIDE 19

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps Let the lexicographic order on [n] × [n] Then implies lexicographic order on V s.t. [n]↑ = degrees, [n]↓ = subscripts

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SLIDE 20

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps Let the lexicographic order on [n] × [n] Then implies lexicographic order on V s.t. [n]↑ = degrees, [n]↓ = subscripts

  • NG(v) denotes the neighbors of v in realization G then

Theorem (Havel’s Lemma, 1955) If H ⊂ V \ {v} and |H| = |NG(v)| and NG(v) H then there exists realization G′ such that NG′(v) = H.

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SLIDE 21

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps Let the lexicographic order on [n] × [n] Then implies lexicographic order on V s.t. [n]↑ = degrees, [n]↓ = subscripts

  • NG(v) denotes the neighbors of v in realization G then

Theorem (Havel’s Lemma, 1955) If H ⊂ V \ {v} and |H| = |NG(v)| and NG(v) H then there exists realization G′ such that NG′(v) = H.

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SLIDE 22

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2a

For complete graphs there are much simpler solutions:

  • Havel A remark on the existence of finite graphs. (Czech),

ˇ Casopis Pˇ

  • est. Mat. 80 (1955), 477–480.

Greedy algorithm to find a realization time complexity O( di) based on swaps Let the lexicographic order on [n] × [n] Then implies lexicographic order on V s.t. [n]↑ = degrees, [n]↓ = subscripts

  • NG(v) denotes the neighbors of v in realization G then

Theorem (Havel’s Lemma, 1955) If H ⊂ V \ {v} and |H| = |NG(v)| and NG(v) H then there exists realization G′ such that NG′(v) = H. there exists canonical realization

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b

Hakimi rediscovered (On the realizability of a set of integers as degrees of

the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506.)

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SLIDE 24

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b

Hakimi rediscovered (On the realizability of a set of integers as degrees of

the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506.)

from that time on it is called Havel-Hakimi algorithm

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SLIDE 25

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b

Hakimi rediscovered (On the realizability of a set of integers as degrees of

the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506.)

from that time on it is called Havel-Hakimi algorithm

  • J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73

(1951), 663–689.

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SLIDE 26

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b

Hakimi rediscovered (On the realizability of a set of integers as degrees of

the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506.)

from that time on it is called Havel-Hakimi algorithm

  • J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73

(1951), 663–689.

all possible graphs with multiple edges but no loops to find all possible molecules with given composition

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SLIDE 27

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b

Hakimi rediscovered (On the realizability of a set of integers as degrees of

the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506.)

from that time on it is called Havel-Hakimi algorithm

  • J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73

(1951), 663–689.

all possible graphs with multiple edges but no loops to find all possible molecules with given composition introduced swaps (but called transfusion)

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SLIDE 28

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b

Hakimi rediscovered (On the realizability of a set of integers as degrees of

the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506.)

from that time on it is called Havel-Hakimi algorithm

  • J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73

(1951), 663–689.

all possible graphs with multiple edges but no loops to find all possible molecules with given composition introduced swaps (but called transfusion) another method: Erd˝

  • s-Gallai theorem (Graphs with prescribed

degree of vertices (in Hungarian), Mat. Lapok 11 (1960), 264–274.)

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SLIDE 29

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Degree sequences 2b

Hakimi rediscovered (On the realizability of a set of integers as degrees of

the vertices of a simple graph. J. SIAM Appl. Math. 10 (1962), 496–506.)

from that time on it is called Havel-Hakimi algorithm

  • J. K. Senior: Partitions and their Representative Graphs, Amer. J. Math., 73

(1951), 663–689.

all possible graphs with multiple edges but no loops to find all possible molecules with given composition introduced swaps (but called transfusion) another method: Erd˝

  • s-Gallai theorem (Graphs with prescribed

degree of vertices (in Hungarian), Mat. Lapok 11 (1960), 264–274.)

used Havel’s theorem in the proof

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one

Let d graphical degree sequence, G and G′ two realizations

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one

Let d graphical degree sequence, G and G′ two realizations looking for a sequence of realizations G1 = H0, H1, . . . , Hk = G2 s.t. ∀i = 0, . . . , k − 1 ∃ swap operation Hi → Hi+1

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one

Let d graphical degree sequence, G and G′ two realizations looking for a sequence of realizations G1 = H0, H1, . . . , Hk = G2 s.t. ∀i = 0, . . . , k − 1 ∃ swap operation Hi → Hi+1 Lemma ∃ swap Hi → Hi+1 then ∃ swap Hi+1 → Hi

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one

Let d graphical degree sequence, G and G′ two realizations looking for a sequence of realizations G1 = H0, H1, . . . , Hk = G2 s.t. ∀i = 0, . . . , k − 1 ∃ swap operation Hi → Hi+1 Lemma ∃ swap Hi → Hi+1 then ∃ swap Hi+1 → Hi by Havel-Hakimi’s lemma such swap-sequence always exists

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one

Let d graphical degree sequence, G and G′ two realizations looking for a sequence of realizations G1 = H0, H1, . . . , Hk = G2 s.t. ∀i = 0, . . . , k − 1 ∃ swap operation Hi → Hi+1 Lemma ∃ swap Hi → Hi+1 then ∃ swap Hi+1 → Hi by Havel-Hakimi’s lemma such swap-sequence always exists for i = 1, 2

  • ∃Gi → canonical realizations
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one

Let d graphical degree sequence, G and G′ two realizations looking for a sequence of realizations G1 = H0, H1, . . . , Hk = G2 s.t. ∀i = 0, . . . , k − 1 ∃ swap operation Hi → Hi+1 Lemma ∃ swap Hi → Hi+1 then ∃ swap Hi+1 → Hi by Havel-Hakimi’s lemma such swap-sequence always exists for i = 1, 2

  • ∃Gi → canonical realizations
  • swap-distance ≤ O( di)
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Transforming one realization into an other one

Let d graphical degree sequence, G and G′ two realizations looking for a sequence of realizations G1 = H0, H1, . . . , Hk = G2 s.t. ∀i = 0, . . . , k − 1 ∃ swap operation Hi → Hi+1 Lemma ∃ swap Hi → Hi+1 then ∃ swap Hi+1 → Hi Theorem (Petersen, 1891 - see Erd˝

  • s-Gallai paper)

by Havel-Hakimi’s lemma such swap-sequence always exists for i = 1, 2

  • ∃Gi → canonical realizations
  • swap-distance ≤ O( di)
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite and directed cases

Erd˝

  • s-Gallai type result for bipartite graphs
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite and directed cases

Erd˝

  • s-Gallai type result for bipartite graphs
  • D. Gale A theorem on flows in networks,

Pacific J. Math. 7 (2) (1957), 1073–1082.

H.J. Ryser Combinatorial properties of matrices of zeros and ones,

  • Canad. J. Math. 9 (1957), 371–377.
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SLIDE 39

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite and directed cases

Erd˝

  • s-Gallai type result for bipartite graphs
  • D. Gale A theorem on flows in networks,

Pacific J. Math. 7 (2) (1957), 1073–1082.

flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones,

  • Canad. J. Math. 9 (1957), 371–377.

binary matrices

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite and directed cases

Erd˝

  • s-Gallai type result for bipartite graphs
  • D. Gale A theorem on flows in networks,

Pacific J. Math. 7 (2) (1957), 1073–1082.

flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones,

  • Canad. J. Math. 9 (1957), 371–377.

binary matrices for both it was byproduct to prove EG-type results for directed graphs: no multiply edges, but possible loops

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite and directed cases

Erd˝

  • s-Gallai type result for bipartite graphs
  • D. Gale A theorem on flows in networks,

Pacific J. Math. 7 (2) (1957), 1073–1082.

flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones,

  • Canad. J. Math. 9 (1957), 371–377.

binary matrices for both it was byproduct to prove EG-type results for directed graphs: no multiply edges, but possible loops Both used bipartite graph representation of directed graphs

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SLIDE 42

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite and directed cases

Erd˝

  • s-Gallai type result for bipartite graphs
  • D. Gale A theorem on flows in networks,

Pacific J. Math. 7 (2) (1957), 1073–1082.

flow theory H.J. Ryser Combinatorial properties of matrices of zeros and ones,

  • Canad. J. Math. 9 (1957), 371–377.

binary matrices for both it was byproduct to prove EG-type results for directed graphs: no multiply edges, but possible loops Both used bipartite graph representation of directed graphs Ryser used swap-sequence transformation from one realization to an other one

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

1

Definitions and History

2

Undirected swap-sequences

3

Bipartite degree sequences

4

Directed degree sequences

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs

G simple graph with red/blue edges - r(v) / b(v) degrees G is balanced : ∀v ∈ V(G) r(v) = b(v).

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs

G simple graph with red/blue edges - r(v) / b(v) degrees G is balanced : ∀v ∈ V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs

G simple graph with red/blue edges - r(v) / b(v) degrees G is balanced : ∀v ∈ V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail Lemma balanced ⇒ E(G) decomposed to alternating circuits

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs

G simple graph with red/blue edges - r(v) / b(v) degrees G is balanced : ∀v ∈ V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail Lemma balanced ⇒ E(G) decomposed to alternating circuits Lemma C = v1, v2, . . . v2n alternating; vi = vj with j − i is even. C can be decomposed into two, shorter alternating circuits.

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs

G simple graph with red/blue edges - r(v) / b(v) degrees G is balanced : ∀v ∈ V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail Lemma balanced ⇒ E(G) decomposed to alternating circuits Lemma C = v1, v2, . . . v2n alternating; vi = vj with j − i is even. C can be decomposed into two, shorter alternating circuits. v v

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs

G simple graph with red/blue edges - r(v) / b(v) degrees G is balanced : ∀v ∈ V(G) r(v) = b(v). trail - no multiple edges circuit - closed trail Lemma balanced ⇒ E(G) decomposed to alternating circuits Lemma C = v1, v2, . . . v2n alternating; vi = vj with j − i is even. C can be decomposed into two, shorter alternating circuits. v v v

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 2

maxCu(G) = # of circuits in a max. circuit decomposition

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 2

maxCu(G) = # of circuits in a max. circuit decomposition circuit C is elementary if

1 no vertex appears more than twice in C, 2 ∃i, j s.t. vi and vj occur only once in C and they have

different parity (their distance is odd).

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 2

maxCu(G) = # of circuits in a max. circuit decomposition circuit C is elementary if

1 no vertex appears more than twice in C, 2 ∃i, j s.t. vi and vj occur only once in C and they have

different parity (their distance is odd). Lemma Let C1, . . . , Cℓ be a max. size circuit decomposition of G. ⇒ each circuit is elementary.

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3

Proof. (i) no vertex occurs 3 times

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3

Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3

Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) ∃ vertex v occurring once - INDIRECT with min. distance

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3

Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) ∃ vertex v occurring once - INDIRECT with min. distance v u1 u2 u1 v

  • dd
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3

Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) ∃ vertex v occurring once - INDIRECT with min. distance v u1 u2 u1 v even

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3

Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) ∃ vertex v occurring once - INDIRECT with min. distance v u1 u2 u1 v even (iv) by pigeon hole: ∃ ≥ 2 vertices occurring once

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Red/blue graphs 3

Proof. (i) no vertex occurs 3 times (ii) when v occurs twice - their distance is odd (iii) ∃ vertex v occurring once - INDIRECT with min. distance v u1 u2 u1 v even (iv) by pigeon hole: ∃ ≥ 2 vertices occurring once (v) by p.h. : ∃u, v occurring once with odd distance

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d,

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SLIDE 61

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E)

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SLIDE 62

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2.

slide-63
SLIDE 63

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
slide-64
SLIDE 64

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
  • ∃v ∈ C occurring once, starting a red edge
slide-65
SLIDE 65

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
  • ∃v ∈ C occurring once, starting a red edge

v u red edges miss stop graph

slide-66
SLIDE 66

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
  • ∃v ∈ C occurring once, starting a red edge

v u red edges miss stop graph if (u, v) ∈ E1, E2

slide-67
SLIDE 67

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
  • ∃v ∈ C occurring once, starting a red edge

v u red edges miss stop graph if (u, v) ∈ E1, E2

slide-68
SLIDE 68

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
  • ∃v ∈ C occurring once, starting a red edge

v u red edges miss stop graph if (u, v) ∈ E1, E2

  • ne swap in start graph

stop graph did not change; new start graph, with sym.

  • diff. having 2 edges less
slide-69
SLIDE 69

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
  • ∃v ∈ C occurring once, starting a red edge

v u red edges miss stop graph if (u, v) ∈ E1, E2

slide-70
SLIDE 70

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
  • ∃v ∈ C occurring once, starting a red edge

v u red edges miss stop graph if (u, v) ∈ E1, E2

  • ne swap in stop graph
slide-71
SLIDE 71

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

One alternating circuit

Realizations G1 and G2 of d, take E1∆E2 = E, and the red/blue graph G = (V, E) Theorem if E(G) is one alternating elementary circuit C of length 2ℓ ⇒ ∃ swap sequence of length ℓ − 1 from G1 to G2. Proof. Gi start (stop) graphs, induction on actual

  • Estart∆Estop
  • ∃v ∈ C occurring once, starting a red edge

v u red edges miss stop graph if (u, v) ∈ E1, E2

  • ne swap in stop graph

start graph did not change; new stop graph, with sym.

  • diff. having 2 edges less
slide-72
SLIDE 72

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Shortest swap sequences in undirected case

G1 and G2 realizations of d. distu(G1, G2) = length of the shortest swap sequence maxCu(G1, G2) = # of circuits in a

  • max. circuit decomposition of E1∆E2
slide-73
SLIDE 73

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Shortest swap sequences in undirected case

G1 and G2 realizations of d. distu(G1, G2) = length of the shortest swap sequence maxCu(G1, G2) = # of circuits in a

  • max. circuit decomposition of E1∆E2

Theorem (Erd˝

  • s-Király-Miklós, 2012)

For all pairs of realizations G1, G2 we have distu(G1, G2) = |E1∆E2| 2 − maxCu(G1, G2).

slide-74
SLIDE 74

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Shortest swap sequences in undirected case

G1 and G2 realizations of d. distu(G1, G2) = length of the shortest swap sequence maxCu(G1, G2) = # of circuits in a

  • max. circuit decomposition of E1∆E2

Theorem (Erd˝

  • s-Király-Miklós, 2012)

For all pairs of realizations G1, G2 we have distu(G1, G2) = |E1∆E2| 2 − maxCu(G1, G2). Very probably the values are NP-complete to be computed

slide-75
SLIDE 75

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Shortest swap sequences in undirected case

G1 and G2 realizations of d. distu(G1, G2) = length of the shortest swap sequence maxCu(G1, G2) = # of circuits in a

  • max. circuit decomposition of E1∆E2

Theorem (Erd˝

  • s-Király-Miklós, 2012)

For all pairs of realizations G1, G2 we have distu(G1, G2) = |E1∆E2| 2 − maxCu(G1, G2). Very probably the values are NP-complete to be computed New upper bound: distu(G1, G2) ≤ |E1∆E2| 2 − 1

slide-76
SLIDE 76

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences?

How to find a typical realization of a degree sequence?

slide-77
SLIDE 77

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences?

How to find a typical realization of a degree sequence?

  • large social networks only # of connections known (PC)
slide-78
SLIDE 78

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences?

How to find a typical realization of a degree sequence?

  • large social networks only # of connections known (PC)
  • online growing network modeling
slide-79
SLIDE 79

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences?

How to find a typical realization of a degree sequence?

  • large social networks only # of connections known (PC)
  • online growing network modeling

huge # of realizations - no way to generate all & choose

slide-80
SLIDE 80

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences?

How to find a typical realization of a degree sequence?

  • large social networks only # of connections known (PC)
  • online growing network modeling

huge # of realizations - no way to generate all & choose Sampling realizations uniformly -

slide-81
SLIDE 81

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences?

How to find a typical realization of a degree sequence?

  • large social networks only # of connections known (PC)
  • online growing network modeling

huge # of realizations - no way to generate all & choose Sampling realizations uniformly - MCMC methods

slide-82
SLIDE 82

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences?

How to find a typical realization of a degree sequence?

  • large social networks only # of connections known (PC)
  • online growing network modeling

huge # of realizations - no way to generate all & choose Sampling realizations uniformly - MCMC methods to estimate mixing time - need to know distances

slide-83
SLIDE 83

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Why a shortest swap-sequences?

How to find a typical realization of a degree sequence?

  • large social networks only # of connections known (PC)
  • online growing network modeling

huge # of realizations - no way to generate all & choose Sampling realizations uniformly - MCMC methods to estimate mixing time - need to know distances distu(G1, G2) ≤ |E1∆E2| 2 ·

  • 1 − 4

3n

  • ≤
  • i

min(di, |V| − di) 1 2 − 2 3n

  • ≤
  • i

di 1 − 4 3n

slide-84
SLIDE 84

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2)

slide-85
SLIDE 85

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ia) and realizations G1 = H0, H1, . . . , Hk−1, Hk = G2 s.t. ∀i realizations Hi and Hi+1 differ exactly in Ci.

slide-86
SLIDE 86

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ia) and realizations G1 = H0, H1, . . . , Hk−1, Hk = G2 s.t. ∀i realizations Hi and Hi+1 differ exactly in Ci.

  • each circuit is elementary
  • for all pairs Hi, Hi+1 the previous theorem is applicable
slide-87
SLIDE 87

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition

slide-88
SLIDE 88

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma ∃ edge in any other circuits which divides C1 into two

  • dd-long trails.
slide-89
SLIDE 89

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma ∃ edge in any other circuits which divides C1 into two

  • dd-long trails.
slide-90
SLIDE 90

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma ∃ edge in any other circuits which divides C1 into two

  • dd-long trails.
slide-91
SLIDE 91

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma ∃ edge in any other circuits which divides C1 into two

  • dd-long trails.
slide-92
SLIDE 92

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma ∃ edge in any other circuits which divides C1 into two

  • dd-long trails.
slide-93
SLIDE 93

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma ∃ edge in any other circuits which divides C1 into two

  • dd-long trails.
slide-94
SLIDE 94

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma ∃ edge in any other circuits which divides C1 into two

  • dd-long trails.

# of circuits unchanged, ∃ shorter circuit - contradiction

slide-95
SLIDE 95

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length

(i) ≤ - take a maximal alternating circuit decomposition C1, ..., CmaxCu(G1,G2) (ib) assume shortest circuit C1 is the shortest among all circuits in all possible minimal circuit decomposition Lemma ∃ edge in any other circuits which divides C1 into two

  • dd-long trails.

1 consider the (actual) symmetric difference, 2 find a maximal circuit decomposition with a shortest

elementary circuit,

3 apply the procedure of one elementary circuit, 4 repeat the whole process with the new (and smaller)

symmetric difference.

slide-96
SLIDE 96

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2).

slide-97
SLIDE 97

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2). G1 = H0, H1, . . . , Hk−1, Hk = G2 minimum real. sequence ∀i the graphs Hi and Hi+1 are in swap-distance 1

slide-98
SLIDE 98

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2). G1 = H0, H1, . . . , Hk−1, Hk = G2 minimum real. sequence ∀i the graphs Hi and Hi+1 are in swap-distance 1 swap subsequence from Hi to Hj also a minimum one

slide-99
SLIDE 99

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2). G1 = H0, H1, . . . , Hk−1, Hk = G2 minimum real. sequence ∀i the graphs Hi and Hi+1 are in swap-distance 1 swap subsequence from Hi to Hj also a minimum one induction on i

slide-100
SLIDE 100

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2). G1 = H0, H1, . . . , Hk−1, Hk = G2 minimum real. sequence ∀i the graphs Hi and Hi+1 are in swap-distance 1 swap subsequence from Hi to Hj also a minimum one induction on i - find circuit decomposition with i circuits: maxCu(G1, Hi) ≥ |E1∆E(Hi)| 2 − distu(G1, Hi)

slide-101
SLIDE 101

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2). G1 = H0, H1, . . . , Hk−1, Hk = G2 minimum real. sequence ∀i the graphs Hi and Hi+1 are in swap-distance 1 swap subsequence from Hi to Hj also a minimum one induction on i - find circuit decomposition with i circuits: maxCu(G1, Hi) ≥ |E1∆E(Hi)| 2 − distu(G1, Hi) analyze the intersection of E(Hi)∆E(Hi+1) with E1∆E(Hi)

slide-102
SLIDE 102

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2). G1 = H0, H1, . . . , Hk−1, Hk = G2 minimum real. sequence ∀i the graphs Hi and Hi+1 are in swap-distance 1 swap subsequence from Hi to Hj also a minimum one induction on i - find circuit decomposition with i circuits: maxCu(G1, Hi) ≥ |E1∆E(Hi)| 2 − distu(G1, Hi) analyze the intersection of E(Hi)∆E(Hi+1) with E1∆E(Hi) empty

slide-103
SLIDE 103

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2). G1 = H0, H1, . . . , Hk−1, Hk = G2 minimum real. sequence ∀i the graphs Hi and Hi+1 are in swap-distance 1 swap subsequence from Hi to Hj also a minimum one induction on i - find circuit decomposition with i circuits: maxCu(G1, Hi) ≥ |E1∆E(Hi)| 2 − distu(G1, Hi) analyze the intersection of E(Hi)∆E(Hi+1) with E1∆E(Hi) empty not empty

slide-104
SLIDE 104

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Proof of shortest swap sequences length 2

(ii) LHS ≥ RHS - we realign the inequality: maxCu(G1, G2) ≥ |E1∆E2| 2 − distu(G1, G2). G1 = H0, H1, . . . , Hk−1, Hk = G2 minimum real. sequence ∀i the graphs Hi and Hi+1 are in swap-distance 1 swap subsequence from Hi to Hj also a minimum one induction on i - find circuit decomposition with i circuits: maxCu(G1, Hi) ≥ |E1∆E(Hi)| 2 − distu(G1, Hi) analyze the intersection of E(Hi)∆E(Hi+1) with E1∆E(Hi) empty not empty

slide-105
SLIDE 105

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

1

Definitions and History

2

Undirected swap-sequences

3

Bipartite degree sequences

4

Directed degree sequences

slide-106
SLIDE 106

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite degree sequences

G(U, V; E) simple bipartite graph, bipartite degree sequence: (ℓ ≤ k) bd(G) =

  • a1, . . . , ak
  • ,
  • b1, . . . , bℓ
  • ,
slide-107
SLIDE 107

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite degree sequences

G(U, V; E) simple bipartite graph, bipartite degree sequence: (ℓ ≤ k) bd(G) =

  • a1, . . . , ak
  • ,
  • b1, . . . , bℓ
  • ,

everything goes through - but be careful - f.e. with swap

slide-108
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite degree sequences

G(U, V; E) simple bipartite graph, bipartite degree sequence: (ℓ ≤ k) bd(G) =

  • a1, . . . , ak
  • ,
  • b1, . . . , bℓ
  • ,

everything goes through - but be careful - f.e. with swap maximum circuit decomposition = set of elementary cycles

slide-109
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite degree sequences

G(U, V; E) simple bipartite graph, bipartite degree sequence: (ℓ ≤ k) bd(G) =

  • a1, . . . , ak
  • ,
  • b1, . . . , bℓ
  • ,

everything goes through - but be careful - f.e. with swap maximum circuit decomposition = set of elementary cycles the cycles can be processed in an arbitrary order

slide-110
SLIDE 110

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Bipartite degree sequences

G(U, V; E) simple bipartite graph, bipartite degree sequence: (ℓ ≤ k) bd(G) =

  • a1, . . . , ak
  • ,
  • b1, . . . , bℓ
  • ,

everything goes through - but be careful - f.e. with swap maximum circuit decomposition = set of elementary cycles the cycles can be processed in an arbitrary order distu(B1, B2) ≤ |E(B1)∆E(B2)| 2 · ℓ − 1 ℓ ≤ 2

  • i

min

  • ai, ℓ − ai
  • 1

2 − 1 2ℓ

  • ≤
  • i

ai

  • ℓ − 1

ℓ .

slide-111
SLIDE 111

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

1

Definitions and History

2

Undirected swap-sequences

3

Bipartite degree sequences

4

Directed degree sequences

slide-112
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Directed degree sequences

  • G(X;

E) simple directed graph, X = {x1, x2, . . . , xn} dd( G) = d+

1 , d+ 2 , . . . , d+ n

  • ,
  • d−

1 , d− 2 , . . . , d− n

slide-113
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Directed degree sequences

  • G(X;

E) simple directed graph, X = {x1, x2, . . . , xn} dd( G) = d+

1 , d+ 2 , . . . , d+ n

  • ,
  • d−

1 , d− 2 , . . . , d− n

representative bipartite graph B( G) = (U, V; E) (Gale) ui ∈ U - out-edges from vi ∈ V in-edges to xi.

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Directed degree sequences

  • G(X;

E) simple directed graph, X = {x1, x2, . . . , xn} dd( G) = d+

1 , d+ 2 , . . . , d+ n

  • ,
  • d−

1 , d− 2 , . . . , d− n

representative bipartite graph B( G) = (U, V; E) (Gale) ui ∈ U - out-edges from vi ∈ V in-edges to xi. E

  • B(

G1)

  • ∆E(B
  • G2)
  • u1

v2 u3 v4

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Directed degree sequences

  • G(X;

E) simple directed graph, X = {x1, x2, . . . , xn} dd( G) = d+

1 , d+ 2 , . . . , d+ n

  • ,
  • d−

1 , d− 2 , . . . , d− n

representative bipartite graph B( G) = (U, V; E) (Gale) ui ∈ U - out-edges from vi ∈ V in-edges to xi. E

  • B(

G1)

  • ∆E(B
  • G2)
  • u1

v2 u3 v4 E( G1)∆E( G2) x1 x2 x3 x4

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Possible problems in E(B1)∆E(B2)

Goal: apply results on bipartite degree sequences for directed degree sequences.

slide-117
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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Possible problems in E(B1)∆E(B2)

Goal: apply results on bipartite degree sequences for directed degree sequences. there are two problems

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Possible problems in E(B1)∆E(B2)

Goal: apply results on bipartite degree sequences for directed degree sequences. there are two problems ux vy u vz y, z = x where x, y, z ∈ V( G)

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SLIDE 119

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Possible problems in E(B1)∆E(B2)

Goal: apply results on bipartite degree sequences for directed degree sequences. there are two problems ux vy u vz y, z = x where x, y, z ∈ V( G) ua vb uc vx uy vz if a = x ∃ swap in ua, vb, uc, vx

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SLIDE 120

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Possible problems in E(B1)∆E(B2)

Goal: apply results on bipartite degree sequences for directed degree sequences. there are two problems ux vy u vz y, z = x where x, y, z ∈ V( G) ua vb uc vx uy vz if a = x ∃ swap in ua, vb, uc, vx if b = y ∃ swap in vb, uc, vx, uy

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C

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SLIDE 122

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C ua vb uc vx uy vz

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C ua vb uc vx uy vz if - - - is not an edge

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SLIDE 124

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C ua vb uc vx uy vz

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SLIDE 125

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 ux vy uz va = vx ub vc if - - - is not an edge

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 ux vy uz va = vx ub vc

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SLIDE 128

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6

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SLIDE 129

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6 u1 v5 u2 v1 u3 v2 u4 v3 u5 v4

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SLIDE 130

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6 u1 v5 u2 v1 u3 v2 u4 v3 u5 v4

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SLIDE 131

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6 all these swaps are C4-swaps

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SLIDE 132

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6 all these swaps are C4-swaps (iv) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| = 6

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6 all these swaps are C4-swaps (iv) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| = 6 ux vy uz vx uy vz

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SLIDE 134

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6 all these swaps are C4-swaps (iv) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| = 6 ux vy uz vx uy vz

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SLIDE 135

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6 all these swaps are C4-swaps (iv) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| = 6 ux vy uz vx uy vz

triangular C6-swap

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Handling elementary circuits(=cycles)

(i) ∃ua ∈ C s.t. va ∈ C (ii) if ∀x : ux ∈ C ⇔ vx ∈ C but ∃x : |vx − ux| = 3 (iii) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| > 6 all these swaps are C4-swaps (iv) if ∀x : if ui = ux ∈ C and vi+1 = vxC but |C| = 6 ux vy uz vx uy vz

triangular C6-swap

triangular C6-swaps can be necessary

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

b c a d e

  • G1 and

G1 on common set of vertices

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

b c a d e

  • G1 and

G1 on common set of vertices

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SLIDE 139

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

b c a d e

  • G1 and

G1 on common set of vertices ua vb uc va ub vc vd ue ud ve E

  • B(

G1)

  • ∆E
  • B(

G1)

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

there are two different alternating cycle decompositions

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

there are two different alternating cycle decompositions ua vb uc va ub vc va ua ud ve vd ue

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

there are two different alternating cycle decompositions ua vb uc va ud ve ua vb uc va ud ve

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

Whenever a triangular C6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C6

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

Whenever a triangular C6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C6 Lemma maximum size C of E1∆E2 having minimum # triangular C6 Then no triangular C6 kisses any other cycle.

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

Whenever a triangular C6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C6 Lemma maximum size C of E1∆E2 having minimum # triangular C6 Then no triangular C6 kisses any other cycle. weighted swap distance weight(C4-swap) = 1;

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

Whenever a triangular C6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C6 Lemma maximum size C of E1∆E2 having minimum # triangular C6 Then no triangular C6 kisses any other cycle. weighted swap distance weight(C4-swap) = 1; weight(triangular C6-swap) = 2

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Analyzing triangular C6

Whenever a triangular C6 kisses another elementary cycle in the decomposition, they can be re-decomposed without triangular C6 Lemma maximum size C of E1∆E2 having minimum # triangular C6 Then no triangular C6 kisses any other cycle. weighted swap distance weight(C4-swap) = 1; weight(triangular C6-swap) = 2 Theorem Let dd be a directed degree sequence with G1 and G2

  • realizations. Then

distd( G1, G2) = |E1∆E2| 2 − maxCd(G1, G2).

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

M.Drew LaMar’s result

Directed 3-Cycle Anchored Digraphs And Their Application In The Uniform Sampling Of Realizations From A Fixed Degree Sequence, in ACM 2011 Winter Simulation Conference (2011), 1–12.

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

M.Drew LaMar’s result

Directed 3-Cycle Anchored Digraphs And Their Application In The Uniform Sampling Of Realizations From A Fixed Degree Sequence, in ACM 2011 Winter Simulation Conference (2011), 1–12. Theorem Each directed degree sequence realization can be transformed into another one with C4- and triangular C6-swaps

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

M.Drew LaMar’s result

Directed 3-Cycle Anchored Digraphs And Their Application In The Uniform Sampling Of Realizations From A Fixed Degree Sequence, in ACM 2011 Winter Simulation Conference (2011), 1–12. Theorem Each directed degree sequence realization can be transformed into another one with C4- and triangular C6-swaps

  • allowing all C6-swaps with weight 2 we have

Theorem distd( G1, G2) can be achieved with C4- and triangular C6-swaps only

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Catherine Greenhill’s result

A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arXiv 1105.0457v4 (2011), 1–48.

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Catherine Greenhill’s result

A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arXiv 1105.0457v4 (2011), 1–48. Theorem for regular DD sequences C4-swaps only are sufficient

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SLIDE 153

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Catherine Greenhill’s result

A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arXiv 1105.0457v4 (2011), 1–48. Theorem for regular DD sequences C4-swaps only are sufficient a b c d e g a b c d e g

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SLIDE 154

Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Catherine Greenhill’s result

A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arXiv 1105.0457v4 (2011), 1–48. Theorem for regular DD sequences C4-swaps only are sufficient a b c d e g a b c d e g swap sequence generated by Greenhill cannot be a minimal

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Swap- distances P .L. Erdös Definitions and History Undirected swap- sequences Bipartite degree sequences Directed degree sequences

Catherine Greenhill’s result

A polynomial bound on the mixing time of a Markov chain for sampling regular directed graphs, arXiv 1105.0457v4 (2011), 1–48. Theorem for regular DD sequences C4-swaps only are sufficient a b c d e g a b c d e g swap sequence generated by Greenhill cannot be a minimal

  • f course this was never a requirement