Relating the total domination number and the annihilation number of cactus graphs and block graphs

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.

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

“…If t = 2, then G is a unicyclic graph. So G is a cactus graph and by the validity of Conjecture 1.1 for cacti due to Bujtás and Jakovac [3] we have γ t (G) ≤ a(G) + 1. Asuume in the rest that t ∈ {3, 4}.…”

“…Since it was proved in [2] that if the minimum degree δ(G) of G is at least 3, then γ t (G) ≤ ⌊ n(G) 2 ⌋ (see also [20] for this result and [10] for its generalization), Conjecture 1.1 holds for graphs G with minimum degree δ(G) ≥ 3. In the seminal paper [7], the conjecture was verified for trees, while recently Bujtás and Jakovac verified it for cactus graphs and for block graphs [3]. In [22], the conjecture was further verified for the so-called C-disjoint graphs and for generalized theta graphs.…”

On an annihilation number conjecture

“…The relation between the annihilation number and various parameters of a graph were studied in [1,2,7,8,9,10,12,14,15,19,32,33].…”