Changing of the domination number of a graph: edge multisubdivision and edge removal

TL;DR

This paper investigates how the domination number with respect to a property $\

Contribution

It provides necessary and sufficient conditions for changes in the domination number after subdividing edges, and bounds the multisubdivision number for graphs.

Findings

Subdivision of an edge affects the domination number based on edge removal properties.

The domination multisubdivision number is at most 3 for hereditary properties.

Conditions are established for when subdividing an edge changes the domination number.

Abstract

For a graphical property $\mathcal{P}$ and a graph $G$, a subset $S$ of vertices of $G$ is a $\mathcal{P}$-set if the subgraph induced by $S$ has the property $\mathcal{P}$. The domination number with respect to the property $\mathcal{P}$, denoted by $\gamma_{\mathcal{P}} (G)$, is the minimum cardinality of a dominating $\mathcal{P}$-set. We define the domination multisubdivision number with respect to $\mathcal{P}$,denoted by $msd_{\mathcal{P}}(G)$, as a minimum positive integer $k$ such that there exists an edge which must be subdivided $k$ times to change $\gamma_\mathcal{P} (G)$. In this paper (a) we present necessary and sufficient conditions for a change of $\gamma_{\mathcal{P}}(G)$ after subdividing an edge of $G$ once, (b) we prove that if $e$ is an edge of a graph $G$ then $\gamma_\mathcal{P} (G_{e,1}) < \gamma_\mathcal{P} (G)$ if and only if $\gamma_\mathcal{P} (G-e) < \gamma_\mathcal{P} (G)$ ($G_{e,t}$ denote the graph obtained from $G$ by subdivision of $e$ with $t$ vertices), (c) we also prove that for every edge of a graph $G$ is fulfilled $\gamma_{\mathcal{P}}(G-e) \leq \gamma_{\mathcal{P}}(G_{e,3}) \leq \gamma_{\mathcal{P}}(G-e) + 1$, and (d) we show that $msd_{\mathcal{P}}(G) \leq 3$, where $\mathcal{P}$ is hereditary and closed under union with $K_1$.