Some variants of perfect graphs related to the matching number, the vertex cover and the weakly connected domination number

Abstract

Given two types of graph theoretical parameters ρ and σ, we say that a graph G is (σ,ρ)-perfect if σ(H)=ρ(H) for every non-trivial connected induced subgraph H of G. In this work we characterize (γw,τ)-perfect graphs, (γw,α′)-perfect graphs, and (α′,τ)-perfect graphs, where γw(G), τ(G) and α′(G) denote the weakly connected domination number, the vertex cover number and the matching number of G, respectively. Moreover, we give conditions on a graph to have equalities between these three parameters.