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

TL;DR

This paper proves the conjecture relating total domination number and annihilation number for quasi-trees and verifies it for several graph constructions, expanding the classes of graphs where the relationship holds.

Contribution

The paper establishes that the conjecture holds for quasi-trees and verifies it for various graph constructions, including bijection graphs and Mycielskians.

Findings

The conjecture $ ext{γ}_t(G) ext{ ≤ } a(G) + 1$ holds for quasi-trees.

The conjecture is verified for bijection graphs, Mycielskians, and universally-identifying graphs.

The results extend the classes of graphs satisfying the conjecture.

Abstract

The total domination number $\gamma_{t}(G)$ of a graph $G$ is the cardinality of a smallest set $D\subseteq V(G)$ such that each vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ of $G$ is the largest integer $k$ such that there exist $k$ different vertices in $G$ with the degree sum at most $m(G)$. It is conjectured that $\gamma_{t}(G)\leq a(G)+1$ holds for every nontrivial connected graph $G$. The conjecture has been proved for graphs with minimum degree at least $3$, trees, certain tree-like graphs, block graphs, and cactus graphs. In the main result of this paper it is proved that the conjecture holds for quasi-trees. The conjecture is verified also for some graph constructions including bijection graphs, Mycielskians, and the newly introduced universally-identifying graphs.