Unilateral Orientation of Mixed Graphs Tamara Mchedlidze, Antonios - - PowerPoint PPT Presentation

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Unilateral Orientation of Mixed Graphs Tamara Mchedlidze, Antonios Symvonis Dept. of Mathematics, National Technical University of Athens, Athens, Greece. { mchet,symvonis } @math.ntua.gr January 25, 2010 T. Mchedlidze, A. Symvonis Dept. of

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SLIDE 1

Unilateral Orientation of Mixed Graphs

Tamara Mchedlidze, Antonios Symvonis

  • Dept. of Mathematics, National Technical University of Athens, Athens, Greece.

{mchet,symvonis}@math.ntua.gr

January 25, 2010

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 2

Definitions

Strong Digraph Unilateral Digraph

3 4 2 3 1 4 2 1

Strong digraph Unilateral digraph

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 3

Definitions

Mixed Graph Strong Orientation Unilateral Orientation

Unilateral orientation of G Mixed graph G Strong orientation of G

v v v v v v v v v v v v2

4 3 1 4 2 3 3 2 4 1 1

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 4

Definitions

Problem: Given a mixed graph G, determine if G has a strong or a unilateral orientation.

Unilateral orientation of G Mixed graph G Strong orientation of G

v v v v v v v v v v v v2

4 3 1 4 2 3 3 2 4 1 1

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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Some known results

Theorem (Robbins, 1939) A connected graph G has a strongly connected orientation if and

  • nly if G has no bridge.

Theorem (Boesch and Tindell, 1980) A mixed multigraph M admits a strong orientation if and only if M is strong and the underlying multigraph of M is bridgeless.

3 11 8 10 2 9 7 5 1 4 6

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 6

Some known results

Theorem (Chartrand, Harary, Schultz, Wall, 1994) A connected graph G has a unilateral orientation if and only if all

  • f the bridges of G lie on a common path.

Question: What about unilateral orientations of mixed graphs?

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  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 7

Some known results

Theorem (Chartrand, Harary, Schultz, Wall, 1994) A connected graph G has a unilateral orientation if and only if all

  • f the bridges of G lie on a common path.

Question: What about unilateral orientations of mixed graphs? Open till now!

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  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 8

Main Results

Characterization of Unilaterally Orientable Mixed Graphs Linear Time Algorithm Testing if a Given Mixed Graph has a Unilateral Orientation

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 9

Characterization of unilaterally orienable mixed graphs

(b) (a)

2 1 31 33 25 28 26 2 19 20 17 15 12 1 32 30 29 27 24 23 18 16 14 13 11 8 7 10 9 6 5 4 3 2 22 3 1 21 3

M M M

M

m m m Strong component digraph of M

Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ the strong component digraph of M, SC(M), has a hamiltonian path.

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 10

Characterization of unilaterally orienable mixed graphs

Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ strong component digraph of M, SC(M), has a hamiltonian path.

SC(M) without hamiltonian path Stong component digraph by a directed path M

u v a b V U

Vertices a and b are not connected

Sketch of proof:

1 Assume that there is no hamiltonian path in SC(M) 2 SC(M) is acyclic digraph

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 11

Characterization of unilaterally orienable mixed graphs

Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ strong component digraph of M, SC(M), has a hamiltonian path.

SC(M) without hamiltonian path Stong component digraph by a directed path M

u v a b V U

Vertices a and b are not connected

Sketch of proof:

3 There are two vertices in SC(M), u and v, that are not connected by a directed path in either direction

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 12

Characterization of unilaterally orienable mixed graphs

Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ strong component digraph of M, SC(M), has a hamiltonian path.

SC(M) without hamiltonian path Stong component digraph by a directed path M

u v a b V U

Vertices a and b are not connected

Sketch of proof:

4 Corresponding strong components of the biorientation of M, U and V contain vertices a and b that are not connected by a directed path.

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 13

Characterization of unilaterally orienable mixed graphs

Lemma (First necessary condition) If a mixed graph M admits a unilateral orientation ⇒ strong component digraph of M, SC(M), has a hamiltonian path.

SC(M) without hamiltonian path Stong component digraph by a directed path M

u v a b V U

Vertices a and b are not connected

Sketch of proof:

5 M do not admit a unilateral orientation.

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 14

Characterization of unilaterally orienable mixed graphs

(a) (b)

21 2 3 4 5 6 9 10 7 8 11 13 14 16 18 23 24 27 29 30 32 1 12 15 17 20 19 26 28 25 33 31 3,2 1 2 3 3,1 2,3 2,2 2,1 1,3 1,1 3,2 1,1 3,1 22 2,3 1,2 2,2 1,2 2,1 1,3

m m m m m m m m m m m m m m M M M

M

m Bridge graphs Bridgeless−component mixed graph: BC(M) m

Lemma (Second necessary condition) If a mixed graph M admits a unilateral orientation ⇒ the bridge graph, B(M), of each of its strong component is a path.

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 15

Characterization of unilaterally orienable mixed graphs

(a) (b)

21 2 3 4 5 6 9 10 7 8 11 13 14 16 18 23 24 27 29 30 32 1 12 15 17 20 19 26 28 25 33 31 3,2 1 2 3 3,1 2,3 2,2 2,1 1,3 1,1 3,2 1,1 3,1 22 2,3 1,2 2,2 1,2 2,1 1,3

m m m m m m m m m m m m m m M M M

M

m Bridge graphs Bridgeless−component mixed graph: BC(M) m

Theorem (Main result) A mixed graph M admits a unilateral orientation ⇔ the bridgeless-component mixed graph, BC(M), admits a hamiltonian orientation.

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 16

Characterization of unilaterally orienable mixed graphs

(a) (b)

21 2 3 4 5 6 9 10 7 8 11 13 14 16 18 23 24 27 29 30 32 1 12 15 17 20 19 26 28 25 33 31 3,2 1 2 3 3,1 2,3 2,2 2,1 1,3 1,1 3,2 1,1 3,1 22 2,3 1,2 2,2 1,2 2,1 1,3

m m m m m m m m m m m m m m M M M

M

m Bridge graphs Bridgeless−component mixed graph: BC(M) m

Sketch of proof:

(M unilaterally orientable ⇒ BC(M) has a hamiltonian orientation)

Unilateral orientation of M contains a spanning walk C C induces a hamiltonian path on BC(M)

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 17

Characterization of unilaterally orienable mixed graphs

(a) (b)

21 2 3 4 5 6 9 10 7 8 11 13 14 16 18 23 24 27 29 30 32 1 12 15 17 20 19 26 28 25 33 31 3,2 1 2 3 3,1 2,3 2,2 2,1 1,3 1,1 3,2 1,1 3,1 22 2,3 1,2 2,2 1,2 2,1 1,3

m m m m m m m m m m m m m m M M M

M

m Bridge graphs Bridgeless−component mixed graph: BC(M) m

Sketch of proof:

(M unilaterally orientable ⇐ BC(M) has a hamiltonian orientation)

Each vertex of BC(M) admits a strong orientation That together with hamiltonian orientation of BC(M) gives a unilateral orientation

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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SLIDE 18

Characterization of unilaterally orienable mixed graphs

(a) (b)

21 2 3 4 5 6 9 10 7 8 11 13 14 16 18 23 24 27 29 30 32 1 12 15 17 20 19 26 28 25 33 31 3,2 1 2 3 3,1 2,3 2,2 2,1 1,3 1,1 3,2 1,1 3,1 22 2,3 1,2 2,2 1,2 2,1 1,3

m m m m m m m m m m m m m m M M M

M

m Bridge graphs Bridgeless−component mixed graph: BC(M) m

Sketch of proof:

(M unilaterally orientable ⇐ BC(M) has a hamiltonian orientation)

Each vertex of BC(M) admits a strong orientation That together with hamiltonian orientation of BC(M) gives a unilateral orientation

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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Summerizing

Theorem Given a mixed graph M = (V , A, E) , we can decide whether M admits a unilateral orientation in O(V + A + E) time. Moreover, if M is unilaterally orientable, a unilateral orientation can be computed in O(V + A + E) time.

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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Thank You!

  • T. Mchedlidze, A. Symvonis
  • Dept. of Mathematics, National Technical University of Athens.

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