On the fixed number of graphs

A set of vertices S of a graph G is called a fixing set of G‎, ‎if only the trivial automorphism of G fixes every vertex in S‎. ‎The fixing number of a graph is the smallest cardinality of a fixing‎ ‎set‎. ‎The fixed number of a graph G is the minimum k‎, ‎such that ‎every k-set of vertices of G is a fixing set of G‎. ‎A graph G‎ ‎is called a k-fixed graph‎, ‎if its fixing number and fixed number‎ ‎are both k‎. ‎In this paper‎, ‎we study the fixed number of a graph‎ ‎and give a construction of a graph of higher fixed number from a‎ ‎graph of lower fixed number‎. ‎We find the bound on k in terms of‎ ‎the diameter d of a distance-transitive k-fixed graph‎.