Proof of a conjecture on irredundance perfect graphs

Abstract

Let ir(G) and (G) be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H ) ¼ (H ), for every induced subgraph H of G. In this article we present a result which immediately implies three known conjectures on irredundance perfect graphs. ß

A disproof of Henning's conjecture on irredundance perfect graphs

“…Moreover, Theorem 4 implies Theorems 1, 2 and 3. [18,19]) If a graph G does not contain the graphs P 6 and G 1 , G 4 in Fig.1 as induced subgraphs, then G is an irredundance perfect graph.…”

“…Conjecture 3 (Puech [16]) Every (P 6 , G 3 , G 4 )-free graph is irredundance perfect. [20]. Laskar and Pfaff also obtained a number of interesting results on irredundance perfect graphs.…”

“…There are a lot of interesting results on irredundance perfect graphs [2,3,7,10,11,14,15,18], see Section 2 for a short summary. The problem of characterizing irredundance perfect graphs in terms forbidden induced subgraphs was posed by Henning [11] who noted that such a characterization is hard to obtain.…”