On the Fractional fixing number of graphs

TL;DR

This paper introduces a fractional version of the fixing number of a graph, formulates it as a linear programming problem, and characterizes graphs with specific fractional fixing numbers.

Contribution

It defines the fractional fixing number, connects it to linear programming, and characterizes graphs with fractional fixing number equal to half the vertices.

Findings

Fractional fixing number is formulated via linear programming.

Graphs with fractional fixing number |V(G)|/2 are characterized.

Fractional fixing numbers for certain graph families are determined.

Abstract

An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that preserve adjacency and non adjacency of vertices in a graph. A fixing set of a graph $G$ is a subset of vertices of $G$ such that only the trivial automorphism fixes every vertex in $S$. Minimum cardinality of a fixing set of $G$ is called the fixing number of $G$. In this article, we define a fractional version of the fixing number of a graph. We formulate the problem of finding the fixing number of a graph as an integer programming problem. It is shown that a relaxation of this problem leads to a linear programming problem and hence to a fractional version of the fixing number of a graph. We also characterize the graphs $G$ with the fractional fixing number $\frac{|V(G)|}{2}$ and the fractional fixing number of some families of graphs is also obtained.