Even Pairs in Bull-reducible Graphs

“…Also, up to now, there is no polynomial time algorithm to find even pairs in perfectly contractile graphs. In 2006, for example, Figueiredo, Maffray and Maciel proved that every bull-reducible Berge graph or its complement contains an even pair [9], but they do not provide a polynomial time algorithm to find it.…”

Section: Introductionmentioning

confidence: 99%

Even pairs in planar Berge graphs

Sales¹,

Sampaio²

2010

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An even pair in a graph is a pair of vertices such that every induced path between them has even length. It is known that deciding the existence of an even pair in a graph is co-NP-complete [3]. In 1990, Reed conjectured that this problem is polynomial time solvable for perfect graphs [22]. In 2005, Chudnovsky et al. obtained an O(n 9 ) time algorithm to recognize perfect graphs [4]. As a direct consequence, they obtained an O(n 11 ) time algorithm to decide if a perfect graph contains an even pair, proving Reed's conjecure. Even when restricted to planar perfect graphs, this was also the best known algorithm. In this paper, we characterize even pairs in planar perfect graphs. Our characterization leads to an O(n 3 ) time algorithm to decide the existence of an even pair (and to find it if it exists) in this class.

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“…By Theorems 1.4 and 1.1, we know that quasi-parity graphs are perfect. Quasi-parity graphs graphs include every strict quasi parity graphs, and also other classes of graphs: bull-free Berge graphs [13], bull-reducible Berge graphs [14].…”

Section: Introductionmentioning

confidence: 99%