Connectivity and Hyperbolicity of a Graph Nicolas Nisse 1 David - - PowerPoint PPT Presentation

Journ´ ees Graphes et Algorithmes 2014 1/15

Connectivity and Hyperbolicity of a Graph

David Coudert 1 Guillaume Ducoffe 1

Nicolas Nisse 1

1COATI (CNRS, UNS, Inria)

Journ´ ees Graphes et Algorithmes 2014 2/15

Preliminaries

• graphs in this study: simple, unweighted, connected (but possibly infinite)
• we focus on: the relations between two tree-likeness parameters.

Topology: how close is the structure of the graph from a tree ? Geometry: how close is the metric of a graph from a tree ?

Journ´ ees Graphes et Algorithmes 2014 2/15

Preliminaries

• graphs in this study: simple, unweighted, connected (but possibly infinite)
• we focus on: the relations between two tree-likeness parameters.

Topology: how close is the structure of the graph from a tree ? − → treewidth, connectivity Geometry: how close is the metric of a graph from a tree ? − → treelength, hyperbolicity introduced for: routing and distance schemes [Chepoi2008,Kleinberg2007], comparison of phylogenetic networks [Chakerian2010], design of approximation algorithms [Chepoi2007]

Journ´ ees Graphes et Algorithmes 2014 3/15

Topology vs. Geometry

• a unifying approach through tree-decompositions:

T = (TG, W ), TG is a tree and ∀t ∈ V (TG), Wt ⊆ V (G) is a bag. Any edge is contained in (at least) one bag. All bags containing the same vertex u ∈ V (G) induce a subtree of TG.

Journ´ ees Graphes et Algorithmes 2014 3/15

Topology vs. Geometry

• a unifying approach through tree-decompositions:

T = (TG, W ), TG is a tree and ∀t ∈ V (TG), Wt ⊆ V (G) is a bag. Any edge is contained in (at least) one bag. All bags containing the same vertex u ∈ V (G) induce a subtree of TG. − → Topology: minimize the size of bags (= tw(G) + 1) tw(G) = 1 ⇐ ⇒ G is a tree.

Journ´ ees Graphes et Algorithmes 2014 3/15

Topology vs. Geometry

• a unifying approach through tree-decompositions:

T = (TG, W ), TG is a tree and ∀t ∈ V (TG), Wt ⊆ V (G) is a bag. Any edge is contained in (at least) one bag. All bags containing the same vertex u ∈ V (G) induce a subtree of TG. − → Topology: minimize the size of bags (= tw(G) + 1) tw(G) = 1 ⇐ ⇒ G is a tree. distance = minimum number of edges in a path diameter = maximum distance in a subset − → Geometry: minimize the diameter of bags (= tl(G)) tl(G) = 1 ⇐ ⇒ G is a chordal graph (no induced cycle of length > 3).

Journ´ ees Graphes et Algorithmes 2014 4/15

Gromov Hyperbolicity

• defined in any metric space (not necessarily a shortest-path metric)

Definition (X, d) is a tree metric ⇐ ⇒ ∃ T an edge-weighted tree with X ⊆ V (T).

• Hyperbolicity ∼ “how close is the metric space from a tree metric ?”

δ(G) = 0 ⇐ ⇒ G is a block-graph.

Journ´ ees Graphes et Algorithmes 2014 4/15

Gromov Hyperbolicity

• defined in any metric space (not necessarily a shortest-path metric)

Definition (X, d) is a tree metric ⇐ ⇒ ∃ T an edge-weighted tree with X ⊆ V (T).

• Hyperbolicity ∼ “how close is the metric space from a tree metric ?”

− → δ-thin triangles

Journ´ ees Graphes et Algorithmes 2014 4/15

Gromov Hyperbolicity

• defined in any metric space (not necessarily a shortest-path metric)

Definition (X, d) is a tree metric ⇐ ⇒ ∃ T an edge-weighted tree with X ⊆ V (T).

• Hyperbolicity ∼ “how close is the metric space from a tree metric ?”

− → 4-points Condition, [Gromov1987] δ(G) = 1 2 max

u,x,v,y (d(u, v) + d(x, y) − max{d(u, x) + d(v, y), d(u, y) + d(v, x)}) .

Journ´ ees Graphes et Algorithmes 2014 5/15

Examples and first relations

• Complete graph Kn: tw(G) = n − 1 ; tl(G) = 1 ; δ(G) = 0

Journ´ ees Graphes et Algorithmes 2014 5/15

Examples and first relations

• Complete graph Kn: tw(G) = n − 1 ; tl(G) = 1 ; δ(G) = 0
• Cycle Cn: tw(G) = 2; tl(G) =

˚ n

3

ˇ ; δ(G) ∼ ¨ n

4

˝

Journ´ ees Graphes et Algorithmes 2014 5/15

Examples and first relations

• Complete graph Kn: tw(G) = n − 1 ; tl(G) = 1 ; δ(G) = 0
• Cycle Cn: tw(G) = 2; tl(G) =

˚ n

3

ˇ ; δ(G) ∼ ¨ n

4

˝

• Square grid Gn,n: tw(G) = n; tl(G) = δ(G) = n − 1

Journ´ ees Graphes et Algorithmes 2014 5/15

Examples and first relations

• Complete graph Kn: tw(G) = n − 1 ; tl(G) = 1 ; δ(G) = 0
• Cycle Cn: tw(G) = 2; tl(G) =

˚ n

3

ˇ ; δ(G) ∼ ¨ n

4

˝

• Square grid Gn,n: tw(G) = n; tl(G) = δ(G) = n − 1

− → treewidth and treelength (resp. hyperbolicity) are uncomparable. − → treelength and hyperbolicity are comparable [Chepoi2008]: δ(G) ≤ tl(G) ≤ 2δ(G) log n

Journ´ ees Graphes et Algorithmes 2014 6/15

Problems

• When are treewidth and treelength comparable ?
• Can we (upper- or lower-) bound the ratio tl(G)/tw(G) ?
• Can we improve the bounds between tl(G) and δ(G) ?

Journ´ ees Graphes et Algorithmes 2014 7/15

Related work

• tw(G) = O(tl(G)) if G is planar [Dieng2009]
• δ(G) ≤

j ch(G)

2

k 2

and tl(G) ≤ j

ch(G) 2

k , with ch(G) the chordality of the graph [Wu2011].

Journ´ ees Graphes et Algorithmes 2014 8/15

Our contributions

• if G has a distance preserving elimination ordering, then tl(G) ≤ 2tw(G).

(comprise weakly modular graphs, triangle-free tandem-win graphs, cobipartite graphs, etc. . . )

• if G is dismantable, then tl(G) ≤ tw(G).
• if G is δ-hyperbolic, then tl(G) = O(δ · tw(G)).

− → tl(G)/tw(G) = O(δ(G)) in general.

Journ´ ees Graphes et Algorithmes 2014 8/15

Our contributions

• Generic relations between graph hyperbolicity and the hyperbolicity of bags

in an arbitrary tree-decomposition. − → relations between graph hyperbolicity and the hyperbolicity of k-connected components (new preprocessing schemes for graph hyperbolicity). − → (1 + O(tw(G))-approximation of graph hyperbolicity.

Journ´ ees Graphes et Algorithmes 2014 9/15

Method

• upper-bounding the diameter of minimal separators

S is a separator if G \ S is disconnected. S is a minimal separator if ∃ two c.c. A, B of G \ S s.t. N(A) = N(B) = S.

• tree-decomposition ∼ triangulation of the graph

Any minimal triangulation can be obtained by completing all sets of a maximal set of pairwise parallel minimal separators of G [ParraScheffler 1997] − → upper-bound c · |S| on diamG(S) = ⇒ tl(G) ≤ c · tw(G).

Journ´ ees Graphes et Algorithmes 2014 10/15

Distance-preserving elimination ordering

• H is an isometric subgraph if ∀u, v ∈ V (H), dH(u, v) = dG(u, v).

G admits a distance-preserving elimination ordering if ∃v1, . . . , vi, . . . s.t. ∀i, G \ {v1, . . . , vi} is an isometric subgraph.

• Particular cases:

domination ordering: ∀i, ∃j > i s.t. N(vi) \ {v1, . . . , vi−1} ⊆ N[vj]. (hereditary modular graphs [Chepoi1988]) dismantling ordering: ∀i, ∃j > i s.t. N[vi] \ {v1, . . . , vi−1} ⊆ N[vj] (chordal graphs, bridged graphs, etc. . . )

Journ´ ees Graphes et Algorithmes 2014 11/15

Our results

• if G has a distance-elimination ordering, then every finite minimal separator

induces a connected subgraph of the square graph G 2. − → diamG(S) ≤ 2(|S| − 1).

• if G is dismantable, then every finite minimal separator induces a connected

subgraph of G. (extends a result from [Jiang2003] for bridged graphs) − → diamG(S) ≤ |S| − 1.

• using G is δ-hyperbolic =

⇒ G 4δ is dismantable [Chalopin2011] if G is δ-hyperbolic, then every finite minimal separator induces a connected subgraph of some graph power G O(δ). − → diamG(S) = O(δ · (|S| − 1)). All upper-bounds are sharp.

Journ´ ees Graphes et Algorithmes 2014 12/15

Application: Graph hyperbolicity and bags of a tree-decomposition

Theorem Let T = (TG, W ) be an arbitrary tree-decomposition. We have: max

t∈V (TG ) δ(Wt, dG) ≤ δ(G) ≤

max

t∈V (TG ) δ(Wt, dG) + 2 ·

max

{t,t′}∈E(TG ) diamG(Wt ∩ Wt′)

• if ∀{t, t′} ∈ E(TG) we have |Wt ∩ Wt′| ≤ k (k-connected components)

δ(G) ≤ max

t∈V (TG ) δ(Wt, dG) + 2 ·

max

{t,t′}∈E(TG ) diamG(Wt ∩ Wt′) ≤ O(k · δ(G))

• if T is an optimal tree-decomposition

δ(G) ≤ max

t∈V (TG ) δ(Wt, dG) + 2 ·

max

{t,t′}∈E(TG ) diamG(Wt ∩ Wt′) ≤ O(tw(G) · δ(G))

Journ´ ees Graphes et Algorithmes 2014 13/15

Conclusion

• New lower-bounds on treewidth for a large class of graphs

(computing the treewidth is NP-hard in this class, treelength can be approximated up to a constant-factor)

• Graphs can be embedded into an edge-weighted tree with additive stretch

O(δ(G) · min{tw(G), log n}).

• New algorithms for bounded-treewidth graphs in this class
• The first general bridge between structural and metric graph invariants

Journ´ ees Graphes et Algorithmes 2014 14/15

Main open questions

• We have tl(G)/tw(G) = O(δ(G)). Can we find a lower-bound (for instance,

using the genus) ?

• What is the complexity of deciding whether a graph admits a

distance-preserving elimination ordering ?

Journ´ ees Graphes et Algorithmes 2014 15/15