Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2021; 58(1): 171-181

Online first article December 28, 2020      Printed January 31, 2021

https://doi.org/10.4134/BKMS.b200166

Copyright © The Korean Mathematical Society.

On the fixing number of functigraphs

Muhammad Fazil, Imran Javaid, Muhammad Murtaza

Bahauddin Zakariya University Multan; Bahauddin Zakariya University Multan; Bahauddin Zakariya University Multan

Abstract

The fixing number of a graph $G$ is the smallest order of a subset $S$ of its vertex set $V(G)$ such that the stabilizer of $S$ in $G$, $\Gamma_{S}(G)$ is trivial. Let $G_{1}$ and $G_{2}$ be the 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 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.

Keywords: Fixing set, fixing number, functigraph, complete graph, tree, join graph

MSC numbers: 05C25