On the Distinguishing number of Functigraphs

TL;DR

This paper investigates the distinguishing number of functigraphs derived from a base graph, establishing bounds and analyzing specific cases like complete and join graphs to extend understanding of graph symmetries.

Contribution

It introduces the concept of the distinguishing number for functigraphs and provides bounds and specific case analyses, extending prior graph symmetry studies.

Findings

Established sharp lower and upper bounds for the distinguishing number of functigraphs.

Analyzed the distinguishing number for functigraphs of complete graphs.

Studied the behavior of the distinguishing number in join graphs.

Abstract

Let $G_{1}$ and $G_{2}$ be disjoint copies of a graph $G$, and let $g:V(G_{1})\rightarrow V(G_{2})$ be a function. A functigraph $F_{G}$ consists of the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})\cup \{uv:g(u)=v\}$. In this paper, we extend the study of the distinguishing number of a graph to its functigraph. We discuss the behavior of the distinguishing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.