A constructive characterization of total domination vertex critical graphs

Let G be a graph of order n and maximum degree ∆(G) and let γt (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 γt -critical. For any γt -critical graph G, it can be shown that n ≤ ∆(G)(γt (G) − 1) + 1. In this paper, we prove that: Let G be a connected γt -critical graph of order n (n ≥ 3), then n = ∆(G)(γt (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 = ∅, the subgraph induced by A is 1-regular, and every vertex in V (G)− A− {v} has exactly one neighbor in A. c © 2008 Elsevier B.V. All rights reserved.