Locating-Dominating Sets of Functigraphs

TL;DR

This paper investigates the location-domination number in functigraphs derived from a graph and its disjoint copy, establishing bounds and exact values for specific graph classes and functions.

Contribution

It introduces bounds and exact formulas for the location-domination number of functigraphs, expanding understanding of domination parameters in complex graph constructions.

Findings

Established sharp lower and upper bounds for the location-domination number in functigraphs.

Derived exact values for complete graphs under various functions f.

Analyzed the parameter for spanning subgraphs of complete graphs.

Abstract

A locating-dominating set of a graph $G$ is a dominating set of $G$ such that every vertex of $G$ outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of $G$ is the minimum cardinality of a locating-dominating set in $G$. Let $G_{1}$ and $G_{2}$ be the disjoint copies of a graph $G$ and $f:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F^f_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup \{uv:v=f(u)\}$. In this paper, we study the variation of the location-domination number in passing from $G$ to $F^f_{G}$ and find its sharp lower and upper bounds. We also study the location-domination number of functigraphs of the complete graphs for all possible definitions of the function $f$. We also obtain the location-domination number of functigraphs of a family of spanning subgraph of the complete graphs.