DUAL-SERVER TOTAL DOMINATION AND TOTAL DUAL-SERVER DOMINATION

A set S subset of V of vertices in a graph G = (V, E) is dominating set of G if every vertex in V \ S has a neighbor in S. If, in addition, every vertex in S also has a neighbor in S, then S is a total dominating set of G. A set S subset of V is a dual-server dominating set if S can be partitioned into two subsets R and B such that every vertex in V \ S is adjacent to at least one vertex in R and at least one vertex in B. In this paper, we introduce and study two distinct definitions for the total version of dual-server domination. Specifically, let S be a dual-server dominating set with a partition {R, B} of S. If every vertex in S has a neighbor in S, then S is called a dual-server total dominating set; while if every vertex in R has a neighbor in R and every vertex in B has a neighbor in B, then S is called a total dual-server dominating set.