On fixing sets of composition and corona product of graphs

TL;DR

This paper investigates the fixing number, which measures how many vertices need to be labeled to eliminate all automorphisms, for the composition and corona products of graphs, providing bounds and exact formulas under various conditions.

Contribution

It establishes new bounds and exact formulas for the fixing number of composition and corona products of graphs, extending understanding of graph automorphisms in these complex structures.

Findings

Bounds for fix(G_1[G_2]) with connected G_1 and arbitrary G_2

Exact fix(G_1 ⊙ G_2) for non-asymmetric graphs

Maximum fix(G_1 ⊙ G_2) determined by fix(G_1) and fix(G_2)

Abstract

A fixing set $\mathcal{F}$ of a graph $G$ is a set of those vertices of the graph $G$ which when assigned distinct labels removes all the automorphisms from the graph except the trivial one. The fixing number of a graph $G$, denoted by $fix(G)$, is the smallest cardinality of a fixing set of $G$. In this paper, we study the fixing number of composition product, $G_1[G_2]$ and corona product, $G_1 \odot G_2$ of two graphs $G_1$ and $G_2$ with orders $m$ and $n$ respectively. We show that for a connected graph $G_1$ and an arbitrary graph $G_2$ having $l\geq 1$ components $G_2^1$, $G_2^2$, ... $G_2^l,$ $mn-1\geq fix(G_1[G_2])\geq m\left(\sum \limits_{i=1}^{l} fix(G_2^i )\right)$. For a connected graph $G_1$ and an arbitrary graph $G_2$, which are not asymmetric, we prove that $fix(G_1\odot G_2)=m fix( G_2)$. Further, for an arbitrary connected graph $G_{1}$ and an arbitrary graph $G_{2}$ we show that $fix(G_1\odot G_2)= max\{fix(G_1), m fix(G_2)\}$.