On the structure of (pan, even hole)‐free graphs

Abstract

A hole is a chordless cycle with at least four vertices. A pan is a graph that consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our ( )-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our ( 2.5 + )-time algorithm to optimally color them.Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 time the clique number.

The class of (P7,C4,C5)‐free graphs: Decomposition, algorithms, and χ‐boundedness

“…This implies that x is complete to one of B C , 1 1 and is mixed on the other; by symmetry, we may assume that x is complete to B 1 and mixed on C 1 . This proves (5). This proves (5).…”

“…It is well known that P 4 -free graphs are perfect [26]. Finally, we remark that Gaspers and Huang [14] showed that every P C ( , ) 6 4 -free graph G satisfies ≤ ⌊ ⌋ χ G ω G ( ) (3/2) ( ) , and Chudnovsky and Sivaraman [6] proved that every P C ( , ) 5 5 -free graph G satisfies ≤ χ G ( ) 2 ω G ( )−1 . In contrast, since there exist graphs of arbitrarily large girth and chromatic number [10], the class of C C ( , ) 4 5 -free graphs is not χ -bounded.…”

“…This proves(5). By symmetry, we may assume that s s > + , and it follows that ℓ u 1 is complete to X 2 , but is not complete to X 0 (consequently, ∉ is an induced P 7 in R, a contradiction.…”

“…The argument is exactly the same as that in Theorem 4.13 except we use a linear-time algorithm developed in [2] to find a maximum weight clique for proper circular-arc graphs instead of the coloring algorithm in [27]. In [5], it was shown that a (pan, even hole)-free graph either has a clique-cutset or is a clique or is the join of a possibly empty clique and a unit circular-arc graph. A similar result was shown for a related class of graphs called (pan, even hole)-free graphs.…”

“…Furthermore, if a graph G has K maximal cliques, they can all be found in O Kn ( ) 3 time by combining results from [25,29]. Thus, all maximal cliques of a C 4 -free graph can be found in O n ( ) 5 time, and it follows that a maximum weight clique of a C 4 -free graph can be found in O n ( ) 5 time. As mentioned above, we…”