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The total domination number $\gamma_{t}(G)$ of a graph $G$ is the cardinalityof a smallest set $D\subseteq V(G)$ such that each vertex of $G$ has a neighborin $D$. The annihilation number $a(G)$ of $G$ is the largest integer $k$ suchthat there exist $k$ different vertices in $G$ with the degree sum at most$m(G)$. It is conjectured that $\gamma_{t}(G)\leq a(G)+1$ holds for everynontrivial connected graph $G$. The conjecture has been proved for graphs withminimum degree at least $3$, trees, certain tree-like graphs, block graphs, andcactus graphs. In the main result of this paper it is proved that theconjecture holds for quasi-trees. The conjecture is verified also for somegraph constructions including bijection graphs, Mycielskians, and the newlyintroduced universally-identifying graphs.

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs