A note on domination and independence-domination numbers of graphs

Vizing's 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 (mod 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) >= 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.