A constructive characterization of total domination vertex critical graphs

Abstract

Let G be a graph of order n and maximum degree Δ ( G ) and let γ t ( G ) denote the minimum cardinality of a total dominating set of a graph G . A graph G with no isolated vertex is the 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 γ t -critical. For any γ t -critical graph G , it can be shown that n ≤ Δ ( G ) ( γ t ( G ) − 1 ) + 1 . In this paper, we prove that: Let G be a connected γ t -critical graph of order n ( n ≥ 3 ), then n = Δ ( G ) ( γ t ( G ) − 1 ) + 1 if and only if G is regular and, for each v ∈ V ( G ) , there is an A ⊆ V ( G ) − { v } such that N ( v ) ∩ A = 0̸ , the subgraph induced by A is 1-regular, and every vertex in V ( G ) − A − { v } has exactly one neighbor in A .