Settling the total domination-annihilation conjecture for graphs with minimum degree two
The total domination number $\gamma_t(G)$ of a graph $G$ is the minimum cardinality of a set $D\subseteq V(G)$ such that every vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ is the largest integer $k$ for which the sum of the $k$ smallest degrees of $G$ is at most $|E(G)|$. A well-known conjecture, originating from Graffiti.pc and later formulated explicitly by Desormeaux, Haynes, and Henning, asserts that $\gamma_t(G)\le a(G)+1$ for every connected nontrivial graph $G$. The conjecture is known for graphs of minimum degree at least three and for several classes of graphs having vertices of degree one or two. In this paper we settle the minimum-degree-two case. More precisely, we prove $\gamma_t(G)\le a(G)+1$ for every connected graph $G$ with $\delta(G)=2$. The proof combines two sharp bounds on the total domination number with an estimate for the annihilation number. Moreover, in some specific cases, the stronger inequality $\gamma_t(G)\le a(G)$ holds.