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.
Muhammad Fazil, Imran Javaid, Muhammad Murtaza
Bahauddin Zakariya University Multan; Bahauddin Zakariya University Multan; Bahauddin Zakariya University Multan
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
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