Outer-Connected Semitotal Domination in Graphs

Abstract

In this paper, we introduce and initiate the study of outer-connected semitotal domination in graphs. Given a graph G without isolated vertices, a set S of vertices of G is a semitotal dominating set if every vertex outside of S is adjacent to a vertex in S and every vertex in S is of distance at most 2 units from another vertex in S. A semitotal dominating set S of G is an outer-connected semitotal dominating set if either S = V (G) or S ̸= V (G) satisfying the property that the subgraph induced by V (G) \ S is connected. The smallest cardinality ̃γt2(G) of an outer-connected semitotal dominating set is the outer-connected semitotal domination number of
G. First, we determine the specific values of ̃γt2(G) for some special graphs and characterize graphs G for specific (small) values of ̃γt2(G). Finally, we investigate the outer-connected semitotal dominating sets in the join, corona, and composition of graphs and, as a consequence, we determine their corresponding outer-connected semitotal domination numbers.