A note on extremal total domination edge critical graphs

Abstract

A set S ⊆ V(G) is a total dominating set if every vertex in V(G) is adjacent to some vertex in S. The smallest cardinality of any total dominating set is the total domination number γ_t,(G). A graph G is said to be total domination edge critical if γ_t,(G + e) < γ_t,(G) for each edge e ∈ E(G^c). We study the size of certain minimal total domination edge critical graphs and their relation to maximal diameter 2-critical graphs - graphs of diameter 2 whose diameter increases upon the removal of any edge.