On the Fractional fixing number of graphs

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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 |V(G)|/2 and the fractional fixing number of some families of graphs is also obtained.