Perfect graphs of strong domination and independent strong domination

Abstract

Let $\text{γ(G)}$, $\text{i(G)}$, $\text{γ}{}_{\mathrm{<mtext>S</mtext>}}\text{(G)}$ and $\text{i}{}_{\mathrm{<mtext>S</mtext>}}\text{(G)}$ denote the domination number, the independent domination number, the strong domination number and the independent strong domination number of a graph $\text{G}$, respectively. A graph $\text{G}$ is called $\text{γi}$-perfect $\text{(}$domination perfect$\text{)}$ if $\text{γ(H)=i(H)}$, for every induced subgraph $\text{H}$ of $\text{G}$. The classes of $\text{γγ}{}_{\mathrm{<mtext>S</mtext>}}$-perfect, $\text{γ}{}_{\mathrm{<mtext>S</mtext>}}\text{i}{}_{\mathrm{<mtext>S</mtext>}}$-perfect, $\text{ii}{}_{\mathrm{<mtext>S</mtext>}}$-perfect and $\text{γi}{}_{\mathrm{<mtext>S</mtext>}}$-perfect graphs are defined analogously. In this paper we present a number of characterization results on the above classes of graphs. For example, characterizations of $\text{K}{}_4$-free $\text{γ}{}_{\mathrm{<mtext>S</mtext>}}\text{i}{}_{\mathrm{<mtext>S</mtext>}}$-perfect graphs and triangle-free $\text{γi}{}_{\mathrm{<mtext>S</mtext>}}$-perfect graphs are given. Moreover, the strong dominating set and independent strong dominating set problems as well as the weak dominating set and independent weak dominating set problems are shown to be NP-complete on a class of graphs. Several problems and conjectures are proposed.