A note on domination and independence-domination numbers of graphs

Vizing\u27s conjecture is true for graphs ▫$G$▫ satisfying ▫$gamma^i(G) = gamma(G)$▫, where ▫$gamma(G)$▫ is the domination number of a graph ▫$G$▫ and ▫$gamma^i(G)$▫ is the independence-domination number of ▫$G$▫, that is, the maximum, over all independent sets ▫$I$▫ in ▫$G$▫, of the minimum number of vertices needed to dominate ▫$I$▫. The equality ▫$gamma^i(G) = gamma(G)$▫ is known to hold for all chordal graphs and for chordless cycles of length ▫$0 pmod{3}$▫. We prove some results related to graphs for which the above equality holds. More specifically, we show that the problems of determining whether ▫$gamma^i(G) = gamma(G) = 2$▫ and of verifying whether ▫$gamma^i(G) ge 2$▫ are NP-complete, even if ▫$G$▫ is weakly chordal. We also initiate the study of the equality ▫$gamma^i = gamma$▫ in the context of hereditary graph classes and exhibit two infinite families of graphs for which ▫$gamma^i < gamma$▫