paper

Locating-Dominating Sets of Functigraphs

arXiv:1709.05152

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 $f : V (G_{1}) \to V (G_{2})$ be a function. A functigraph $F_{G}^{f}$ 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 $F_{G}^{f}$ 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.

14 pages, 3 figures