A constructive characterization of total domination vertex critical graphs

Let GG be a graph of order nn and maximum degree Δ(G)Δ(G) and let γt(G)γt(G) denote the minimum cardinality of a total dominating set of a graph GG. A graph GG with no isolated vertex is the total domination vertex critical if for any vertex vv of GG that is not adjacent to a vertex of degree one, the total domination number of G−vG−v is less than the total domination number of GG. We call these graphs γtγt-critical. For any γtγt-critical graph GG, it can be shown that n≤Δ(G)(γt(G)−1)+1n≤Δ(G)(γt(G)−1)+1. In this paper, we prove that: Let GG be a connected γtγt-critical graph of order nn (n≥3n≥3), then n=Δ(G)(γt(G)−1)+1n=Δ(G)(γt(G)−1)+1 if and only if GG is regular and, for each v∈V(G)v∈V(G), there is an A⊆V(G)−{v}A⊆V(G)−{v} such that N(v)∩A=0̸N(v)∩A=0̸, the subgraph induced by AA is 1-regular, and every vertex in V(G)−A−{v}V(G)−A−{v} has exactly one neighbor in AA.