The perfection and recognition of bull-reducible Berge graphs

Abstract

The recently announced Strong Perfect Graph Theorem states that the class of perfect graphs coincides with the class of graphs containing no induced odd cycle of length at least 5 or the complement of such a cycle. A graph in this second class is called Berge. A bull is a graph with five vertices x, a, b, c, d and five edges xa, xb, ab, ad, bc. A graph is bull-reducible if no vertex is in two bulls. In this paper we give a simple proof that every bull-reducible Berge graph is perfect. Although this result follows directly from the Strong Perfect Graph Theorem, our proof leads to a recognition algorithm for this new class of perfect graphs whose complexity, O(n 6), is much lower than that announced for perfect graphs.

Transitive orientations in bull-reducible Berge graphs

“…If h ∈ W , this is by the definition of T . If h ∈ P ∪ X ∪ Y and t ∈ A this is by (4), (5) and (6). If h ∈ P ∪ X ∪ Y and t ∈ T − Y − A this is by the definition of Y .…”

“…Suppose that u misses v. Consider any path R uv = r 1 -· · ·-r p given by (7), with p odd, r 1 = u, r p = v. Then v-c 1 -c 2 -u-R uv -v is an odd cycle of length at least five, so it must contains a triangle, for otherwise it contains an odd hole. Note that c 1 and c 2 do not see two consecutive vertices on the path R uv , since they are in Z * 2 and by (5). So, in order to have a triangle, there must be a vertex r j that sees both c 1 , c 2 , and so r j ∈ L 1 , and so 3 ≤ j ≤ p − 2.…”

“…A graph G is called bull-reducible if every vertex of G lies in at most one bull of G. Clearly, bull-free graphs are bull-reducible. Everett, de Figueiredo, Klein and Reed [5] proved that every bull-reducible Berge graph is perfect. Although this result now follows directly from the Strong Perfect Graph Theorem [3], the proof given in [5] is much simpler and leads moreover to a polynomial-time recognition algorithm for bull-reducible Berge graphs whose complexity is lower than that given for all Berge graphs in [2].…”

“…Lemma 1 ( [5,9]) Let G be a bull-reducible C 5 -free graph. If G contains a wheel or a double broom then G has a homogeneous set.…”

“…Everett, de Figueiredo, Klein and Reed [5] proved that every bull-reducible Berge graph is perfect. Although this result now follows directly from the Strong Perfect Graph Theorem [3], the proof given in [5] is much simpler and leads moreover to a polynomial-time recognition algorithm for bull-reducible Berge graphs whose complexity is lower than that given for all Berge graphs in [2]. A graph is called weakly chordal (or "weakly triangulated") if it contains no hole and no antihole.…”