On αrγs( k)-perfect graphs

Abstract

For some integer k⩾0 and two graph parameters π and τ, a graph G is called πτ( k)- perfect, if π( H)− τ( H)⩽ k for every induced subgraph H of G. For r⩾1 let α r and γ r denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α 1 γ 1( k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study α r γ s ( k)-perfect graphs for r, s⩾1. We prove several properties of minimal α r γ s ( k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α 2 r−1 γ r ( k)-perfect graphs for r⩾1. Furthermore, we characterize claw-free α 2 γ 2( k)-perfect graphs.