Satgraphs and independent domination. Part 1

A graph G is called a satgraph if there exists a partition A∪B=V(G) such that
• A induces a clique [possibly, A=∅],
• B induces a matching [i.e., G(B) is a 1-regular subgraph, possibly, B=∅], and
• there are no triangles (a,b,b′), where a∈A and b,b′∈B.We also introduce the hereditary closure of SAT, denoted by HSAT [hereditary satgraphs]. The class HSAT contains split graphs. In turn, HSAT is contained in the class of all (1,2)-split graphs [A. Gyárfás, Generalized split graphs and Ramsey numbers, J. Combin. Theory Ser. A 81 (2) (1998) 255–261], the latter being still not characterized. We characterize satgraphs in terms of forbidden induced subgraphs.There exist close connections between satgraphs and the satisfiability problem [SAT]. In fact, SAT is linear-time equivalent to finding the independent domination number in the corresponding satgraph. It follows that the independent domination problem is NP-complete for the hereditary satgraphs. In particular, it is NP-complete for perfect graphs.