A disproof of Henning's conjecture on irredundance perfect graphs

Abstract

Let ir(G) and γ(G) be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H) = γ(H), for every induced subgraph H of G. In this paper, we disprove the known conjecture of Henning (Chartrand and Lesniak, Graphs & Digraphs, 3rd Edition, Chapman & Hall, London, 1996, p. 321; Henning, Discrete Math. 142 (1995) 107) that a graph G is irredundance perfect if and only if ir(H) = γ(H) for every induced subgraph H of G with ir(H) ≤ 4. We also give a summary of known results on irredundance perfect graphs. Moreover, the irredundant set problem and the dominating set problem are shown to be NP-complete on some classes of graphs. A number of problems and conjectures are proposed. © 2002 Elsevier Science B.V. All rights reserved