On 4-[formula omitted]-critical graphs of diameter two

Abstract

A vertex subset S of graph G is a total dominating set of G if every vertex of G is adjacent to a vertex in S. For a graph G with no isolated vertex, the total domination number of G, denoted by γt(G), is the minimum cardinality of a total dominating set. A total dominating set of cardinality γt(G) is called a γt(G)-set. A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G−v is less than the total domination number of G. We call these graphs γt-critical. If such a graph G has total domination number k, then we call it k-γt-critical. In this note we study 4-γt-critical connected graphs G of diameter two. We prove that such graphs with minimum at least two have order at least 10, and we characterize all 4-γt-critical connected graphs of order 10 with maximum degree 5. Moreover, we obtain some 4-γt-critical connected graphs of order 10 with maximum degree 4 and for any integer k≥2, n=3k+5, there exists a 4-γt-critical graph G of order n with diam(G)=2.