On Metric Dimension of Functigraphs

TL;DR

This paper investigates how the metric dimension, a measure of how uniquely vertices are identified by distances, changes when constructing functigraphs from a base graph, providing bounds and specific cases for complete graphs and cycles.

Contribution

It establishes bounds on the metric dimension of functigraphs derived from connected graphs and explores its behavior on complete graphs and cycles, advancing understanding of this graph invariant.

Findings

Metric dimension of functigraphs is between 2 and 2n-3 for connected graphs of order n.

The paper provides exact metric dimension values for functigraphs on complete graphs.

It analyzes the metric dimension for functigraphs on cycles.

Abstract

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.