TOTAL OUTER-CONNECTED DOMINATION NUMBER OF MIDDLE TREES

Let G = (V, E) be a graph without an isolated vertex. A set D ⊆ V (G) is a total dominating set if D is dominating, and the induced subgraph G[D] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total dominating set of G. A set D ⊆ V (G) is a total outer-connected dominating set if D is total dominating, and the induced subgraph G[V (G) − D] is a connected graph. The total outer-connected domination number of G denoted by γtc(G) is the minimum cardinality of a total outer-connected dominating set of G. In this paper, we study the total outer-connected domination number of middle trees. Indeed, we obtain tight bounds for this number in terms of the order of the graph M(T). We also compute the total outer-connected domination number of some families of graphs such as star graphs, path graphs, spider graphs, and some operation on trees explicitly.