Total domination stability in graphs

A set D of vertices in an isolate-free graph G is a total dominating set of G if every vertex is adjacent to a vertex in D. The total domination number, γ_t(G), of G is the minimum cardinality of a total dominating set of G. We note that γ_t(G)≥2 for every isolate-free graph G. A non-isolating set of vertices in G is a set of vertices whose removal from G produces an isolate-free graph. The γ_t⁻-stability, denoted st_γt⁻(G), of G is the minimum size of a non-isolating set S of vertices in G whose removal decreases the total domination number. We show that if G is a connected graph with maximum degree Δ satisfying γ_t(G)≥3, then st_γt⁻(G)≤2Δ−1, and we characterize the infinite family of trees that achieve equality in this upper bound. The total domination stability, st_γt(G), of G is the minimum size of a non-isolating set of vertices in G whose removal changes the total domination number. We prove that if G is a connected graph with maximum degree Δ satisfying γ_t(G)≥3, then st_γt(G)≤2Δ−1.