The Structure of Graphs Without Even Holes or Odd Pans Kathie - PowerPoint PPT Presentation

The Structure of Graphs Without Even Holes or Odd Pans Kathie Cameron Department of Mathematics Wilfrid Laurier University, Waterloo, Canada Steven Chaplick Institut für Mathematik Technische Universität Berlin Berlin, Germany Chính Hoàng Department of Physics and Computer Science Wilfrid Laurier University, Waterloo, Canada

The Structure of Graphs Without Even Holes or Odd Pans Kathie Cameron Department of Mathematics Wilfrid Laurier University, Waterloo, Canada Steven Chaplick Institut für Mathematik Technische Universität Berlin Berlin, Germany Chính Hoàng Department of Physics and Computer Science Wilfrid Laurier University, Waterloo, Canada

Let G and F be graphs, and F a (possibly infinite) class of graphs. Graph G is F-free if G has no induced subgraph isomorphic to F. Graph G is F - free if G has no induced subgraph isomorphic to any graph in F.

A hole is a chordless cycle with at least least four vertices. C 4 C 5 C 6 A hole is odd or even depending on whether it has an odd or even number of vertices. An antihole is the complement of a hole. _ _ C 5 C 6

There are many interesting classes of graphs that can be characterized as F-free or F -free. Chordal graphs are the hole-free graphs. Berge graphs are the odd-hole-free and odd-antihole-free graphs. Line-graphs claw-free graphs Claw: Information System on Graph Classes http://www.graphclasses.org/ defines even-hole free as (C 6 , C 8 , …) -free. That is, they allow C 4 s. We don’t . They use the term even-cycle-free for (C 4 , C 6 , …) -free.

The structure of even-hole-free graphs is in many ways quite similar to the structure of Berge graphs . Note that by excluding the 4-hole, one also excludes all antiholes of size at least 6. So even-hole-free graphs are more similar to Berge graphs than they are to odd-hole-free graphs.

The structure of even-hole-free graphs is in many ways quite similar to the structure of Berge graphs . Note that by excluding the 4-hole, one also excludes all antiholes of size at least 6. So even-hole-free graphs are more similar to Berge graphs than they are to odd-hole-free graphs.

The class of even-hole-free graphs is in coNP : To show that a graph is not even-hole-free, display an even hole. What can we be shown to make it easy to verify that a given graph IS even-hole-free? Conforti, Cornu é jols, Kapoor, and Vušković (2002) gave a decomposition theorem for even-hole free graphs, which proves that the class is in NP . Complexity of Recognition (n = # vertices, m = # edges) Berge graphs O(n 9 ) Chudnovsky, Cornuejols, Liu, Seymour, Vušković (2005) Even-hole-free graphs O(n 40 ) Conforti, C ornuéjols, Kapoor, Vušković (2002) O(n 31 ) Chudnovsky, Kawarabayashi, Seymour (2005) O(n 19 ) Da Silva, Vušković ( 2013 ) O(n 5 m 3 ) ≤ O(n 11 ) Chang, Liu (2015) Odd-hole-free graphs Unknown

Independent set : a set of vertices, no two of which are joined by an edge Colouring : partition of the vertices into independent sets

Clique : a set of vertices, every pair of which are joined by an edge Clique partition : partition of the vertices into cliques

A simplicial vertex is a vertex whose neighbourhood induces a clique . Theorem (Dirac 1961) . Every chordal graph contains a simplicial vertex.

A simplicial vertex is a vertex whose neighbour-set induces a clique. Theorem . Every chordal graph contains a simplicial vertex. A bisimplicial vertex is a vertex whose neighbour-set is a union of two cliques.

A simplicial vertex is a vertex whose neighbour-set induces a clique. Theorem . Every chordal graph contains a simplicial vertex. A bisimplicial vertex is a vertex whose neighbour-set is a union of two cliques.

A simplicial vertex is a vertex whose neighbour-set induces a clique. Theorem . Every chordal graph contains a simplicial vertex. A bisimplicial vertex is a vertex whose neighbour-set is a union of two cliques. Theorem (Addario-Berry, Chudnovsky, Havet, Reed, Seymour, 2008) Every even-hole-free graph has a bisimplicial vertex.

A simplicial vertex is a vertex whose neighbour-set induces a clique. Theorem . Every chordal graph contains a simplicial vertex. A bisimplicial vertex is a vertex whose neighbour-set is a union of two cliques. Theorem (Addario-Berry, Chudnovsky, Havet, Reed, Seymour, 2008) Every even-hole-free graph has a bisimplicial vertex. This implies that every even-hole-free graph has a vertex which is not the centre of a claw, centre of the claw and suggests that claw-free even-hole-free free graphs may have interesting structure.

A bisimplicial vertex is a vertex whose neighbour-set is a union of two cliques. Theorem (Addario-Berry, Chudnovsky, Havet, Reed, Seymour, 2008) Every even-hole-free graph has a bisimplicial vertex. Corollary: A polytime algorithm for finding a largest clique in an even-hole-free graph.  Let v 1 be a bisimplicial vertex of even-hole-free graph G.    .

A bisimplicial vertex is a vertex whose neighbour-set is a union of two cliques. Theorem (Addario-Berry, Chudnovsky, Havet, Reed, Seymour, 2008) Every even-hole-free graph has a bisimplicial vertex. Corollary: A polytime algorithm for finding a largest clique in an even-hole-free graph.  Let v 1 be a bisimplicial vertex of even-hole-free graph G.  Find the largest clique C 1 in the neighour-set of v 1 .  Either C 1 U { v 1 } is a largest clique in G, or else v 1 is not in a largest clique of G.  Delete v 1 . G - v 1 is even-hole-free. Repeat.

A bisimplicial vertex is a vertex whose neighbour-set is a union of two cliques. Theorem (Addario-Berry, Chudnovsky, Havet, Reed, Seymour, 2008) Every even-hole-free graph has a bisimplicial vertex. ω(G) = size of the largest clique in G χ(G) = minimum number of colours in a colouring of G Corollary (A-B, C, H, R, S, 2008) If G is even-hole- free then χ(G) ≤ 2ω(G) − 1. Proof . Let v be a bisimplicial vertex of even-hole-free graph G. Inductively colour G -v with 2ω(G - v ) −1 colors ≤ 2ω(G) − 1 colours. Since v is bisimplicial, its degree is at most 2ω(G) −2, and hence there is a colour for v among the 2ω(G) − 1 colo urs. □ We prove that: If G is claw-free even-hole-free, then χ(G) ≤ 1.5ω(G )

Largest Clique Largest Minimum Colouring Independent Set Even-Hole polytime unknown unknown Free Graphs Odd-Hole NP-hard unknown NP-hard Free Graphs Claw-Free NP-hard polytime NP-hard Graphs Claw-Free Even-Hole- Free

Largest Clique Largest Minimum Colouring Independent Set Even-Hole- polytime unknown unknown Free Graphs Odd-Hole- NP-hard unknown NP-hard Free Graphs Claw-Free NP-hard polytime NP-hard Graphs Claw-Free polytime polytime ? Even-Hole- Free

Largest Clique Largest Minimum Colouring Independent Set Even-Hole- polytime unknown unknown Free Graphs Odd-Hole- NP-hard unknown NP-hard Free Graphs Claw-Free NP-hard polytime NP-hard Graphs Claw-Free polytime polytime polytime Even-Hole- Free

A pan is a hole together with a pendant edge (the handle). Note that a pan contains a claw. So claw-free graphs are pan-free.

A pan is a hole together with a pendant edge (the handle). Note that a pan contains a claw. So claw-free graphs are pan-free. About Pan-Free Graphs:  Olariu (1989) showed that SPGC held for pan-free graphs  De Simone used the term “apple” for “pan” (1993) and studied largest independent set on a subclass  Largest weight independent set can be found in polytime (Brandstadt, Lozin, Mosca, 2010)  We give a recognition algorithm O(nm 2 )

Largest Clique Largest Minimum Colouring Independent Set Even-Hole- polytime unknown unknown Free Graphs Odd-Hole- NP-hard unknown NP-hard Free Graphs Pan-Free NP-hard polytime NP-hard Graphs Pan-Free polytime polytime polytime Even-Hole- Free

Clique cutset clique G: G 1 C G 2 There are no edges between G 1 – C and G 2 – C

Clique cutset clique G: G 1 C G 2 There are no edges between G 1 – C and G 2 – C Tarjan (1982) and Whitesides (1984) studied algorithmic aspects of clique cutsets.

Where C is a clique-cutset, given optimum colourings of G 1 and G 2 , we can obtain an optimum colouring of G, by identifying the colours on C. (Whitesides 1984) C G 1 G 2 C C

Where C is a clique-cutset, given optimum colourings of G 1 and G 2 , we can obtain an optimum colouring of G, by identifying the colours on C. (Whitesides 1984) C G 1 G 2 C C

Where C is a clique-cutset, given optimum colourings of G 1 and G 2 , we can obtain an optimum colouring of G, by identifying the colours on C. C G 1 G 2 C C

Where C is a clique-cutset, given optimum colourings of G 1 and G 2 , we can obtain an optimum colouring of G, by identifying the colours on C. C G 1 G 2 C C

Clique cutset clique G: G 1 C G 2 G contains an even hole if and only if G 1 or G 2 contains an even hole. G 1 G 2

Clique cutset clique G: G 1 C G 2 G contains an even hole if and only if G 1 or G 2 contains an even hole. G 1 G 2

Clique cutset clique G: G 1 C G 2 G contains an even hole if and only if G 1 or G 2 contains an even hole. G 1 G 2 Not a hole!