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The total domination number and the annihilation number of quasi-trees and composite graphs

Thetotal domination number $\gamma_{t}(G)$ of a graph $G$ is the cardinality of asmallest vertex subset $D$ of $V(G)$ such that each vertex of $G$ has at leastone neighbor in $D$. The annihilation number $a(G)$ of $G$ is the largestinteger $k$ such that there exist $k$ different vertices in $G$ with degree sumof at most the size of $G$. It is conjectured by W. J. Desormeaux et al. that$\gamma_{t}(G)\leq a(G)+1$ holds for every nontrivial connected graph $G$. Theconjecture has been proved for graphs with minimum degree at least 3, trees,tree-like graphs, block graphs and cacti. In this talk, we introduce some ofour results on the above conjecture.

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