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. C4 C5 C6 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. _ _ C5 C6

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.

Claw:

Line-graphs claw-free graphs Information System on Graph Classes http://www.graphclasses.org/ defines even-hole free as (C6, C8, …)-free. That is, they allow C4s. We don’t. They use the term even-cycle-free for (C4, C6, …)-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, Cornuéjols, 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(n9) Chudnovsky, Cornuejols, Liu, Seymour, Vušković (2005) Even-hole-free graphs O(n40) Conforti, Cornuéjols, Kapoor, Vušković (2002)

O(n31) Chudnovsky, Kawarabayashi, Seymour (2005) O(n19) Da Silva, Vušković ( 2013 ) O(n5m3) ≤ O(n11) 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

centre of the claw

has a vertex which is not the centre of a 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 v1 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 v1 be a bisimplicial vertex of even-hole-free graph G.  Find the largest clique C1 in the neighour-set of v1.  Either C1 U {v1} is a largest clique in G, or else v1 is not in a largest clique of G.  Delete v1. G - v1 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 colours. □ 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(nm2)

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: G1 C G2 There are no edges between G1 – C and G2 – C

Clique cutset clique G: G1 C G2 There are no edges between G1 – C and G2 – C Tarjan (1982) and Whitesides (1984) studied algorithmic aspects of clique cutsets.

Where C is a clique-cutset, given optimum colourings of G1 and G2, we can obtain an optimum colouring of G, by identifying the colours on C. (Whitesides 1984) C G1 G2 C C

Where C is a clique-cutset, given optimum colourings of G1 and G2, we can obtain an optimum colouring of G, by identifying the colours on C. (Whitesides 1984) C G1 G2 C C

Where C is a clique-cutset, given optimum colourings of G1 and G2, we can obtain an optimum colouring of G, by identifying the colours on C. C G1 G2 C C

Where C is a clique-cutset, given optimum colourings of G1 and G2, we can obtain an optimum colouring of G, by identifying the colours on C. C G1 G2 C C

Clique cutset clique G: G1 C G2 G contains an even hole if and only if G1 or G2 contains an even hole. G1 G2

Clique cutset clique G: G1 C G2 G contains an even hole if and only if G1 or G2 contains an even hole. G1 G2

Clique cutset clique G: G1 C G2 G contains an even hole if and only if G1 or G2 contains an even hole. G1 G2 Not a hole!

Clique cutset | G clique | G1 G2 G: G1 C G2

Decomposition tree

Clique Cutset Decomposition and DecompositionTree: Where G has clique-cutset C, G is decomposed into G1 and G2. G is the root of the decomposition tree, and G1 and G2 are its children. If Gi has a clique-cutset, then Gi is decomposed, and is the root of subtree representing its decomposition.

C G1 G2

C G1 G2 C1 C2

C G1 G2 C1 C2

G1 G2 C1 C2 Indecomposable atoms

Clique cutset | G clique | G1 G2 G: G1 C G2

Decomposition tree

Clique Cutset Decomposition and DecompositionTree: Where G has clique-cutset C, G is decomposed into G1 and G2. G is the root of the decomposition tree, and G1 and G2 are its children. If Gi has a clique-cutset, it is decomposed, and is the root of subtrees representing its decomposition. Graphs which do not have a clique cutset are called atoms. Tarjan (1982) A clique decomposition tree can be constructed in O(nm) time and the number of leaves of the tree is at most n-1. That is, the number of atoms (indecomposable graphs) produced by the decomposition tree is at most n-1. (n = # vertices, m = #edges)

How to Find a Pan in a Graph G b a For each edge ab, to check if ab is the handle of a pan: Look for a hole containing a in G-N(b). (Also, check if ba is the handle of a pan.) Find a hole in graph F containing vertex a or that no such hole exists Construct in O(nm) time the clique cutset decompostion tree T(F)  If a is universal in all the atoms containing it, then a does not lie on any hole of F  If a is not universal in some atom A,

How to Find a Pan in a Graph G b a For each edge ab, to check if ab is the handle of a pan: Look for a hole containing a in G-N(b). (Also, check if ba is the handle of a pan.) Find a hole in graph F containing vertex a or that no such hole exists Construct in O(nm) time the clique cutset decompostion tree T(F)  If a is universal in all the atoms containing it, then a does not lie on any hole of F  If a is not universal in some atom A, then compute the components of the non-neighbours of a : C1,…,Ct C1 N(a) a C2

How to Find a Pan in a Graph G b a For each edge ab, to check if ab is the handle of a pan: Look for a hole containing a in G-N(b). (Also, check if ba is the handle of a pan.) Find a hole in graph F containing vertex a or that no such hole exists Construct in O(nm) time the clique cutset decompostion tree T(F)  If a is universal in all the atoms containing it, then a does not lie on any hole of F  If a is not universal in some atom A, then compute the components of the non-neighbours of a : C1,…,Ct Let Ni be the set of neighbours of a which have neighbours in Ci C1 N1 a

How to Find a Pan in a Graph G b a For each edge ab, to check if ab is the handle of a pan: Look for a hole containing a in G-N(b). (Also, check if ba is the handle of a pan.) Find a hole in graph F containing vertex a or that no such hole exists Construct in O(nm) time the clique cutset decompostion tree T(F)  If a is universal in all the atoms containing it, then a does not lie on any hole of F  If a is not universal in some atom A, then compute the components of the non-neighbours of a : C1,…,Ct Let Ni be the set of neighbours of a which have neighbours in Ci Ni can not be a clique Choose some Ni , choose two non-adjacent vertices x and y in Ni Find a chordless path from x to y whose interior lies entirely in Ci C1 N1 x y a

How to Find a Pan in a Graph G b a in O(nm2) time For each edge ab, to check if ab is the handle of a pan: Look for a hole containing a in G-N(b). (Also, check if ba is the handle of a pan.) Find a hole in graph F containing vertex a or that no such hole exists Construct in O(nm) time the clique cutset decompostion tree T(F)  If a is universal in all the atoms containing it, then a does not lie on any hole of F  If a is not universal in some atom A, then compute the components of the non-neighbours of a : C1,…,Ct Let Ni be the set of neighbours of a which have neighbours in Ci Ni can not be a clique Choose some Ni , choose two non-adjacent vertices x and y in Ni Find a chordless path from x to y whose interior lies entirely in Ci A hole a containing a linear time

A circular-arc graph is the intersection graph of a set of arcs of a circle. It is called proper if no arc properly contains another. It is called unit if each arc has unit length. Unit CA Proper CA CA

http://en.wikipedia.org/wiki/Circular-arc_graph

Circular-arc graphs and proper and unit circular-arc graphs can be recogized in polytime, and largest independent set, largest clique, and minimum clique partition can be solved in polytime for graphs in these classes. However, finding a minimum colouring is NP-hard for circular-arc graphs but polytime- solvable for proper and unit circular-arc graphs.

A circular-arc graph is the intersection graph of a set of arcs of a circle. It is called proper if no arc properly contains another. It is called unit if each arc has unit length.

http://en.wikipedia.org/wiki/Circular-arc_graph

Proper hence unit circular-arc graphs are pan-free. Our main result: pan-free even-hole-free graphs can be decomposed by clique cutsets into (essentially) unit circular-arc graphs.

The join of two graphs H and J is obtained by adding all edges between them. H J

The join of two graphs H and J is obtained by adding all edges between them. H J

The join of two graphs H and J is obtained by adding all edges between them. H J Decomposition Theorem (C, Chaplick, Hoàng) Let G be a connected pan-free even-hole-free graph. Then: (1) G is a clique (2) G contains a clique-cutset (3) G is a unit circular-arc graph OR (4) G is the join of a unit circular-arc graph and a clique

Assume G is the join of a clique C and another graph H. H C In any colouring of G, each vertex of C must get a colour which is not used for any other vertex. Thus χ(G) = χ(H) + |C|

Pan-Free Even-Hole-Free Decomposition Theorem Let G be a connected pan-free even-hole-free graph. Then: (1) G is a clique (2) G contains a clique-cutset (3) G is a unit circular-arc graph OR (4) G is the join of a unit circular-arc graph and a clique Colouring Algorithm for Pan-Free Even-Hole-Free Graphs  Find the clique cutset decomposition of the graph G The decomposition tree has O(n) leaves (atoms)  By the Decomposition Theorem, each atom is a clique, a unit circular-arc graph or the join of a unit circular-arc graph and a clique.  It is easy to recognize if a graph is a clique or is the join of a clique and another graph. If an atom is not one of these, then it is a unit circular arc graph.  Unit circular-arc graphs can be coloured in O(n1.5+m) time and thus so can the join of a unit circular-arc graph and a clique.  Work up the tree from the leaves to root to colour the whole graph.

Pan-Free Even-Hole-Free Decomposition Theorem Let G be a connected pan-free even-hole-free graph. Then: (1) G is a clique (2) G contains a clique-cutset (3) G is a unit circular-arc graph OR (4) G is the join of a unit circular-arc graph and a clique Colouring Algorithm for Pan-Free Even-Hole-Free Graphs  Find the clique cutset decomposition of the graph G The decomposition tree has O(n) leaves (atoms)  By the Decomposition Theorem, each atom is a clique, a unit circular-arc graph or the join of a unit circular-arc graph and a clique.  It is easy to recognize if a graph is a clique or is the join of a clique and another graph. If an atom is not one of these, then it is a unit circular arc graph.  Unit circular-arc graphs can be coloured in O(n1.5+m) time

[Lin, Szwarcfiter; 2008] [Shih, Hsu;1989]

 Work up the tree from the leaves to root to colour the whole graph.

Recognizing Pan-Free Even-Hole-Free Graphs We can check for the existence of a pan in O(nm2) time. Recall: Where C is a clique cutset clique G: G1 C G2 G contains an even hole if and only if G1 or G2 contains an even hole. Algorithm to Test for an Even Hole in a Pan-Free Graph  Construct the clique-cutset decomposition tree (in O(nm) time)  G has an even hole if and only if one of the O(n) atoms does.  Our decomposition theorem says: Since G is pan-free, if G is even- hole-free, each atom is a clique, a unit circular-arc graph or the join

• f a clique and a unit circular-arc graph. Test this.

 Then we need to test for an even hole in circular-arc graphs O(mnloglogn) [C, Eschen, Hoàng, Sritharan 2007]

Minimum Colouring Recognition Even-Hole- unknown O(n40) Conforti, Cornuéjols, Free Graphs

Kapoor, Vušković (2002)

O(n31) Chudnovsky, Kawarabayashi,

Seymour (2005)

O(n19) Da Silva, Vušković (2013) O(n5m3) Chang, Liu (2012) Odd-Hole- NP-hard unknown Free Graphs Pan-Free NP-hard O(nm2) Graphs Pan-Free O(n2.5+nm) Even-Hole- ≤ O(n3) O(nm2loglogn) by unit CA graphs Free O(nm) by buoy construction

For l ≥ 5, an l-buoy is a collection of cliques, B1, B2, ..., Bl , such that each vertex in Bi has a neighbour in Bi-1 and one in Bi+1 and no neighbours in any of the other set Bj.

For l ≥ 5, an l-buoy is a collection of cliques, B1, B2, ..., Bl , such that each vertex in Bi has a neighbour in Bi-1 and one in Bi+1 and no neighbours in any of the other set Bj.

For l ≥ 5, an l-buoy is a collection of cliques, B1, B2, ..., Bl , such that each vertex in Bi has a neighbour in Bi-1 and one in Bi+1 and no neighbours in any of the other set Bj. The sets Bi are called the bags of the buoy.

Decomposition Theorem (C, Chaplick, Hoàng) Let G be a connected pan-free even-hole-free graph. Then: (1) G is a clique (2) G contains a clique-cutset OR (3) For every maximal buoy B of G, G is the join of B and a (possibly empty) clique

Decomposition Theorem (C, Chaplick, Hoàng) Let G be a connected pan-free even-hole-free graph. Then: (1) G is a clique (2) G contains a clique-cutset OR (3) For every maximal buoy B of G, G is the join of B and a (possibly empty) clique Theorem (C,C,H) If B is a buoy in a C4 –free graph, then B is a circular-arc graph.

Decomposition Theorem (C, Chaplick, Hoàng) Let G be a connected pan-free even-hole-free graph. Then: (1) G is a clique (2) G contains a clique-cutset OR (3) For every maximal buoy B of G, G is the join of B and a (possibly empty) clique Theorem (C,C,H) If B is a buoy in a C4 –free graph, then B is a circular-arc graph. Theorem (C,C,H) If B is a buoy in a pan-free even-hole-free graph, then B is a unit circular-arc graph.

Decomposition Theorem (C, Chaplick, Hoàng) Let G be a connected pan-free even-hole-free graph. Then: (1) G is a clique (2) G contains a clique-cutset OR (3) For every maximal buoy B of G, G is the join of B and a (possibly empty) clique Theorem (C,C,H) If B is a buoy in a pan-free even-hole-free graph, then B is a unit circular-arc graph.

Decomposition Theorem (C, Chaplick, Hoàng) Let G be a connected pan-free even-hole-free graph. Then: (1) G is a clique (2) G contains a clique-cutset (3) G is a unit circular-arc graph OR (4) G is the join of a unit circular-arc graph and a clique Theorem (C,C,H) If B is an buoy in a pan-free even-hole-free graph, then B is a unit circular-arc graph.

Some properties of buoys in (even hole, pan)-free graphs:  the vertices of each bag Bi can be ordered by neighbourhood inclusion  For each i, either (Bi-1 U Bi) or (Bi U Bi+1) is a clique

• Theorem. For any graph G,

either G contains an even hole

• r G contains a pan
• r for every clique-cutset decomposition, each atom is a clique or a

blue/grey induced subgraph with pink neighbour-set). may be empty Totally adjacent No 2K2

EP Theorem. For any graph G, either G contains an even hole

• r G contains a pan
• r there is a clique-cutset decompostion with at most n-1 atoms such

that each atom is either a clique or a blue/grey induced subgraph with pink neighbour-set). may be empty Totally adjacent No 2K2

17

H-Free Graphs Decomposition Theorem Applications

Buoys To Circular Arcs

Remember: each bag is orderable by neighbourhood inclusion.

Bi Bi+1 x2 b1 x1 b2 bt -1

(i,1) (i,2) (i, ) (i) (i+1)

... ... ... bti xti xt -1

i

(i, -1) ti ti

i

... (0) (1)

( -1)

(i+1) (i)

. . . ... ...

{

{

18

H-Free Graphs Decomposition Theorem Applications

Buoys To Unit Circular Arcs

Remember: every other pair of bags is a clique.

Bi Bi+1 x2 b1 x1 b2 bt -1

(i,1) (i,2) (i, ) (i) (i+1)

... ... ... bti xti xt -1

i

(i, -1) ti ti

i

...

{

Ai+

{

Ai

{

Ai+1

(i,1) (i,2)

...

(i, -1) ti

...

{

length=1

{

length=1

{

Ai-1+

(i,1) (i,2)

...

(i, -1) ti

...

{

length=1

{

length=1

{

Ai+1+

(i, ) ti (i, ) ti

length=1

Bi-1 Bi+2

(i-1) (i+2)

Improved Algorithm for Recognizing (Pan, Even Hole)-Free Graphs

• 1. Run Clique Cutset decomposition. There will be ≤ n-1 atoms O(nm)
• 2. For each atom (O(n) of them):

3. Find a hole O(n+m) 4. Build a special buoy from this hole O(n+m) 5. Check the constructed buoy for “bad” subgraphs O(n+m)

• 6. Bottom-up on the decomposition tree to find odd pans O(nm)

Overall: O(nm) Chordality Testing: O(n+m) [Rose, Tarjan, Lueker; 1976]

Are even-hole-free line-graphs interesting?

Are even-hole-free line-graphs interesting? NO.

Are even-hole-free line-graphs interesting? NO. Jack Edmonds pointed out that: If G = L(H) is even-hole free, then H does not contain two edge- disjoint cycles.

Are even-hole-free line-graphs interesting? NO. Jack Edmonds pointed out that: If G = L(H) is even-hole free, then H does not contain two edge- disjoint cycles. So H is a cactus of odd cycles and edges.

Treewidth A tree decomposition of a graph G is a tree T together with a function which maps each vertex v of G to a subtree T(v) of T such that for every edge uv of G, T(u) ∩ T(v) is non-empty The vertices of T are called bags, and the size of a bag b is the number

• f vertices v of G such that b is in T(v).

The treewidth of a tree decomposition is max{|b|-1 : b is a bag of T} The treewidth (tw) of G is the smallest treewidth of any tree decomposition of G. Lemma: If G has a clique cutset C and G1 , …, Gk are the components of G-C, then tw(G) = maxi tw(Gi U C) Theorem: A (pan, even hole)-free graph G has tw(G)+1 ≤ 1.5 ω(G)

• Proof. By the Lemma and our Structure Theorem, we only have to show

the theorem holds for any buoy in G.

Theorem: A (pan, even hole)-free graph G has tw(G)+1 ≤ 1.5 ω(G)

• Proof. By the Lemma and our Structure Theorem, we only have to show

the theorem holds for any buoy in G.

Theorem: A (pan, even hole)-free graph G has tw(G)+1 ≤ 1.5 ω(G)

• Proof. By the Lemma and our Structure Theorem, we only have to show

the theorem holds for any buoy in G. For each bag Bi, either Bi-1U Bi or Bi U Bi+1 is a clique. Cut the buoy at the smallest bag Bk and unroll it onto a line.

Theorem: A (pan, even hole)-free graph G has tw(G)+1 ≤ 1.5 ω(G)

• Proof. By the Lemma and our Structure Theorem, we only have to show

the theorem holds for any buoy in G. For each bag Bi, either Bi-1U Bi or Bi U Bi+1 is a clique. Cut the buoy at the smallest bag Bk and unroll it onto a line.

Theorem: A (pan, even hole)-free graph G has tw(G)+1 ≤ 1.5 ω(G)

• Proof. By the Lemma and our Structure Theorem, we only have to show

the theorem holds for any buoy in G. For each bag Bi, either Bi-1U Bi or Bi U Bi+1 is a clique. Cut the buoy at the smallest bag Bk and unroll it onto a line. This is the tree for the tree decomposition.

Theorem: A (pan, even hole)-free graph G has tw(G)+1 ≤ 1.5 ω(G)

• Proof. By the Lemma and our Structure Theorem, we only have to show

the theorem holds for any buoy in G. For each bag Bi, either Bi-1U Bi or Bi U Bi+1 is a clique. Cut the buoy at the smallest bag Bk and unroll it onto a line. This is the tree for the tree decomposition. Add the vertices of Bk to each bag to get a tree representation of G. □

Theorem: A (pan, even hole)-free graph G has tw(G)+1 ≤ 1.5 ω(G) Observation: For a graph G, χ(G) ≤ tw(G) + 1 Corollary: An (pan, even hole)-free graph G has χ(G) ≤ 1.5 ω(G)

Claw-Free Even-Hole Free Graphs Need not be β-perfect Simple Greedy Colouring Algorithm  Call the colours 1, 2, 3,…  Order the vertices.  Colour the vertices in order, using the smallest colour available. Where Δ(G) is the maximum degree of a vertex of G, this algorithm colours G using at most Δ(G) + 1 colours. We can do somewhat better using a special ordering: Improved Greedy Colouring Algorithm  Call the colours 1, 2, 3,…  Order the vertices by repeatedly removing a vertex of minimum degree in the graph remaining.  Colour the vertices in reverse order, using the smallest colour available.

Improved Greedy Colouring Algorithm  Call the colours 1, 2, 3,…  Order the vertices by repeatedly removing a vertex of minimum degree in the graph remaining.  Colour the vertices in reverse order, using the smallest colour available. Let δ(H) = minimum degree of a vertex of H. χ(G) ≤ the number of colours used to colour G by this algorithm ≤ Maximum{ δ(H) +1 : H is an induced subgraph of G } = β(G) defn Markossian, Gasparian and Reed (1996) defined a graph to be β-Perfect if for every induced subgraph F of G, χ(F) = β(F). Even holes C2k are not β-perfect since χ (C2k) = 2 but β(C2k) = 3. The following example shows that claw-free even-hole-free graphs need not be β-perfect.

The following example shows that claw-free even-hole-free graphs need not be β-perfect.

Claw-free Even-hole-free Diamond-free even-hole-free β – perfect Kloks, Müller, diamond Vušković (20009)

Open Problems

(1) Odd-hole-free graphs  Recognition  Structure Theorem  Largest Independent Set

ω+2 2

Recent result (Scott, Seymour): χ(G) ≤ 2 (2) Even-hole-free graphs  Largest Independent Set  Minimum Colouring (3) Claw-free (or pan-free) even-hole-free Minimum clique partition

(4) Pan-free odd-hole-free We think we can colour them (5) Pan-free Independent Set Polytope Thank you