Changing and unchanging of the domination number of a graph: Path addition numbers

TL;DR

This paper investigates how the domination number of a graph changes when a path is added between two vertices, establishing bounds and conditions for when the domination number increases, decreases, or remains unchanged.

Contribution

It provides new bounds and necessary and sufficient conditions for the domination number's behavior under path addition between vertices.

Findings

   

   

  

Abstract

Given a graph $G = (V,E)$ and two its distinct vertices $u$ and $v$. The $(u,v)$-$P_k$-{\em addition graph} of $G$ is the graph $G_{u,v,k-2}$ obtained from disjoint union of $G$ and a path $P_k: x_0,x_1,..,x_{k-1}$, $k \geq 2$, by identifying the vertices $u$ and $x_0$, and identifying the vertices $v$ and $x_{k-1}$. We prove that (a) $ \gamma(G)-1 \leq \gamma(G_{u,v,k})$ for all $k \geq 1$, and (b) $\gamma(G_{u,v,k}) > \gamma(G)$ when $k \geq 5$. We also provide necessary and sufficient conditions for the equality $\gamma(G_{u,v,k}) = \gamma(G)$ to be valid for each pair $u,v \in V(G)$. pair $u,v \in V(G)$.