Geodesic
Relating the total domination number and the annihilation number of cactus graphs and block graphs
Ars Mathematica Contemporanea, Tome 16 (2019) no. 1, pp. 183-202.

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The total domination number γt(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 γt(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.
DOI : 10.26493/1855-3974.1378.11d
Keywords: Total domination number, annihilation number, cactus graph, block graph 
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Csilla Bujtás; Marko Jakovac. Relating the total domination number and the annihilation number of cactus graphs and block graphs. Ars Mathematica Contemporanea, Tome 16 (2019) no. 1, pp. 183-202. doi : 10.26493/1855-3974.1378.11d. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1378.11d/

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