Relating the annihilation number and the total domination number for some graphs

• Published in: Discrete Applied Mathematics Vol. 332; pp. 41 - 46
• Main Authors: Hua, Xinying, Xu, Kexiang, Hua, Hongbo
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.06.2023
• Subjects: Annihilation number; Complementary prism; Inflated graph of tree; Maximal outerplanar graph; Square graph of tree; Total domination number; Complementary prism; Square graph of tree; Maximal outerplanar graph; Annihilation number; Total domination number; Inflated graph of tree
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2023.01.018

The total domination numberγt(G) of a graph G is the cardinality of a smallest vertex subset D of V(G) such that each vertex of G has at least one neighbor in D. The annihilation numbera(G) of G is the largest integer k such that there exist k different vertices in G with degree sum of at most the size of 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 some graphs with minimum degree 1 or 2. In this paper, we prove that the above conjecture holds for some graphs with minimum degree at most two. More specifically, we prove that the above conjecture holds for inflated graphs of trees, square graphs of trees, maximal outerplanar graphs and complementary prisms of graphs.