On the total outer-connected domination in graphs
A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph induced by V?S is connected. The total outer-connected domination number ? toc (G) is the minimum size of such a set. We give some properties and bounds for ? toc in general graphs and in trees. For graphs of order n, diameter 2 and minimum degree at least 3, we show that \(\gamma_{toc}(G)\le \frac{2n-2}{3}\) and we determine the extremal graphs.
total outer-connected dominating set total outer-connected domination number diameter