The Existence Problem of $$k$$ k - $$\gamma _t$$ γ t -Critical Graphs

Abstract

A set $$S$$ of vertices in graph $$G$$ is a total dominating set of $$G$$ if every vertex of $$G$$ is adjacent to some vertex in $$S$$ . The minimum cardinality of a total dominating set of $$G$$ is the total domination number $$\gamma _t(G)$$ . A graph $$G$$ with no isolated vertex is 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 $$G \backslash \{v\}$$ is less than the total domination number of $$G$$ . We call such graphs $$\gamma _t$$ -critical. If such a graph $$G$$ has total domination number $$k$$ , we call it $$k$$ - $$\gamma _t$$ -critical. It is well known from Sohn et al. (Discrete Appl Math 159:46–52, 2011) that the only remaining open cases are $$\Delta (G) = 5, 7,9$$ for $$k=5$$ and $$\Delta (G) =7,9, 11$$ for $$k=7$$ . In this paper, the existence of $$\Delta (G) = 5, 7$$ for $$k=5$$ are presented and the existence of $$\Delta (G) = 7, 9$$ for $$k=7$$ are presented.