Abstract
The total domination number γ t ( G ) of a graph G is the order of a smallest set D ⊆ V ( G ) such that each vertex of G is adjacent to some vertex in D . The annihilation number a ( G ) of G is the largest integer k such that there exist k different vertices in G with degree sum of at most | E ( G )| . It is conjectured that γ t ( G ) ≤ a ( G ) + 1 holds for every nontrivial connected graph G . The conjecture was proved for graphs with minimum degree at least 3 , and remains unresolved for graphs with minimum degree 1 or 2 . In this paper we establish the conjecture for cactus graphs and block graphs.
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