Some Results on Total Outer-Paired Dominating Sets in Graphs

Let G=(V, E) be a simple graph with vertex set V and edge set E. An outer-paired dominating set D of a graph G is a dominating set such that the subgraph induced by V\D has a perfect matching. The outer-paired domination number of G denoted by is the minimum cardinality of an outer-paired dominating set of G. Moreover, a total outer-paired dominating set D of a graph G is a total dominating set such that the subgraph induced by V\D has a perfect matching. The total outer-paired domination number of G denoted by is the minimum cardinality of a total outer-paired dominating set of G. In this paper, besides introducing such a notion for a graph, we study some fundamental properties of this invariant. Furthermore, we present some bounds in terms of the order, the size, the girth of a graph and etc. Finally, the well-known Nordhaus-Gaddum inequality is obtained for regular graphs.