Domination properties and induced subgraphs (Q686437)

Two types of classes of graphs are studied. The class \(\text{Forb}(C_ t,P_ t)\) is the class of all graphs which contain no induced subgraph isomorphic to the circuit \(C_ t\) with \(t\) vertices or to the path \(P_ t\) with \(t\) vertices. The class \(\text{Dom}(d,k)\) is the class of graphs \(G\) in which every connected induced subgraph \(H\) contains a subgraph \(D\) of diameter at most \(d\) such that the vertex set of \(D\) is \(k\)-dominating in \(H\). Here a \(k\)-dominating set in a graph is the subset of the vertex set such that each vertex not belonging to it has the distance at most \(k\) from some vertex belonging to it. Equalities and inclusions between \(\text{Forb}(C_ t,P_ t)\) and \(\text{Dom}(d,k)\) for various values of \(t\), \(d\) and \(k\) are studied.