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 ( γ w , τ ) -perfect graphs, ( γ w , α ′ )-perfect graphs, and ( α ′ , τ )-perfect graphs, where γ w ( 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.