Secure domination number of $k$-subdivision of graphs

TL;DR

This paper investigates the secure domination number in the context of $k$-subdivisions of graphs, providing new insights into how subdivision affects secure domination properties.

Contribution

It introduces the concept of secure domination number for $k$-subdivisions of graphs and analyzes its behavior, which is a novel extension in domination theory.

Findings

Derived bounds for secure domination number of $k$-subdivided graphs

Characterized secure domination number for specific classes of graphs

Provided algorithms for computing secure domination in subdivided graphs

Abstract

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. A dominating set $D$ is called a secure dominating set of $G$, if for every $u\in V-D$, there exists a vertex $v\in D$ such that $uv \in E$ and $D-\{v\}\cup\{u\}$ is a dominating set of $G$. The cardinality of a smallest secure dominating set of $G$, denoted by $\gamma_s(G)$, is the secure domination number of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the secure domination number of $k$-subdivision of $G$.