Doubly Connected 2-Domination in the Join of Two Graphs

Given a nontrivial connected graph, a set of vertices of a graph is a doubly connected 2-dominating set if the induced subgraphs of the set and its complement are connected, and every vertex in its complement is adjacent to at least two vertices in the set. The doubly connected 2-domination number of a nontrivial connected graph is the cardinality of a minimum doubly connected 2-dominating set. In this paper, we characterize the doubly connected 2-dominating sets in the join of two graphs and de