Special kinds of domination parameters of edge-deleted graphs. (Q2716008)

A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is called dominating in \(G\), if each vertex of \(G\) either is in \(D\), or is adjacent to a vertex of \(D\). The minimum number of vertices of a dominating set in \(G\) is the domination number \(\gamma (G)\) of \(G\). A dominating set \(D\) in \(G\) is called a split dominating set in \(G\), if the subgraph of \(G\) induced by \(V(G)-D\) is disconnected. A dominating set \(D\) in \(G\) is a maximal dominating set of \(G\), if \(V(G)-D\) is not a dominating set of \(G\). The minimum number of vertices of a split (or maximal) dominating set in \(G\) is called the split domination number \(\gamma _s(G)\) of \(G\) (or maximal domination number \(\gamma _m(G)\) of \(G\) respectively). The paper studies the behaviour of the described concepts at removing edges from the graph \(G\).