Journ´ ees Graphes et Algorithmes 2014 1/15 Connectivity and Hyperbolicity of a Graph Nicolas Nisse 1 David Coudert 1 Guillaume Ducoffe 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 = ( T G , W ), T G is a tree and ∀ t ∈ V ( T G ) , W t ⊆ 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 T G .
Journ´ ees Graphes et Algorithmes 2014 3/15 Topology vs. Geometry • a unifying approach through tree-decompositions : T = ( T G , W ), T G is a tree and ∀ t ∈ V ( T G ) , W t ⊆ 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 T G . − → 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 = ( T G , W ), T G is a tree and ∀ t ∈ V ( T G ) , W t ⊆ 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 T G . − → 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 K n : tw ( G ) = n − 1 ; tl ( G ) = 1 ; δ ( G ) = 0
Journ´ ees Graphes et Algorithmes 2014 5/15 Examples and first relations • Complete graph K n : tw ( G ) = n − 1 ; tl ( G ) = 1 ; δ ( G ) = 0 ˚ n ¨ n ˇ ˝ • Cycle C n : tw ( G ) = 2; tl ( G ) = ; δ ( G ) ∼ 3 4
Journ´ ees Graphes et Algorithmes 2014 5/15 Examples and first relations • Complete graph K n : tw ( G ) = n − 1 ; tl ( G ) = 1 ; δ ( G ) = 0 ˚ n ¨ n ˇ ˝ • Cycle C n : tw ( G ) = 2; tl ( G ) = ; δ ( G ) ∼ 3 4 • Square grid G n , n : tw ( G ) = n ; tl ( G ) = δ ( G ) = n − 1
Journ´ ees Graphes et Algorithmes 2014 5/15 Examples and first relations • Complete graph K n : tw ( G ) = n − 1 ; tl ( G ) = 1 ; δ ( G ) = 0 ˚ n ¨ n ˇ ˝ • Cycle C n : tw ( G ) = 2; tl ( G ) = ; δ ( G ) ∼ 3 4 • Square grid G n , 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 ) ≤ 2 δ ( 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] j ch ( G ) k 2 • δ ( G ) ≤ 2 j k ch ( G ) and tl ( G ) ≤ , 2 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 ) ≤ 2 tw ( 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 diam G ( 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 ) , d H ( u , v ) = d G ( u , v ). G admits a distance-preserving elimination ordering if ∃ v 1 , . . . , v i , . . . s.t. ∀ i , G \ { v 1 , . . . , v i } is an isometric subgraph. • Particular cases: domination ordering: ∀ i , ∃ j > i s.t. N ( v i ) \ { v 1 , . . . , v i − 1 } ⊆ N [ v j ]. (hereditary modular graphs [Chepoi1988] ) dismantling ordering: ∀ i , ∃ j > i s.t. N [ v i ] \ { v 1 , . . . , v i − 1 } ⊆ N [ v j ] (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 . − → diam G ( 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) − → diam G ( S ) ≤ | S | − 1. ⇒ G 4 δ is dismantable [Chalopin2011] • using G is δ -hyperbolic = if G is δ -hyperbolic, then every finite minimal separator induces a connected subgraph of some graph power G O ( δ ) . − → diam G ( 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 = ( T G , W ) be an arbitrary tree-decomposition. We have: t ∈ V ( T G ) δ ( W t , d G ) ≤ δ ( G ) ≤ max t ∈ V ( T G ) δ ( W t , d G ) + 2 · max { t , t ′ }∈ E ( T G ) diam G ( W t ∩ W t ′ ) max • if ∀{ t , t ′ } ∈ E ( T G ) we have | W t ∩ W t ′ | ≤ k ( k -connected components) δ ( G ) ≤ t ∈ V ( T G ) δ ( W t , d G ) + 2 · max { t , t ′ }∈ E ( T G ) diam G ( W t ∩ W t ′ ) ≤ O ( k · δ ( G )) max • if T is an optimal tree-decomposition δ ( G ) ≤ t ∈ V ( T G ) δ ( W t , d G ) + 2 · max { t , t ′ }∈ E ( T G ) diam G ( W t ∩ W t ′ ) ≤ O ( tw ( G ) · δ ( G )) max
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
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