On Fixing number of Functigraphs

TL;DR

This paper investigates the fixing number of functigraphs, exploring how it relates to the original graph's fixing number and establishing bounds, with specific results for well-known graph families.

Contribution

It introduces bounds for the fixing number of functigraphs derived from a base graph and analyzes this parameter for specific graph classes.

Findings

Established sharp lower and upper bounds for fixing numbers of functigraphs.

Analyzed fixing numbers for complete graphs, trees, and join graphs.

Provided insights into how the fixing number changes from a graph to its functigraph.

Abstract

The fixing number of a graph $G$ is the order of the smallest subset $S$ of its vertex set $V(G)$ such that stabilizer of $S$ in $G$, $\Gamma_{S}(G)$ is trivial. 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 $E(G_{1})\cup E(G_{2})\cup \{uv:v=g(u)\}$. In this paper, we study the behavior of the fixing number in passing from $G$ to $F_{G}$ and find its sharp lower and upper bounds. We also study the fixing number of functigraphs of some well known families of graphs like complete graphs, trees and join graphs.