On the fixed number of graphs

ýA set of vertices S of a graph G is called a fixing set of Gý, ýif only the trivial automorphism of G fixes every vertex in Sý. ýThe fixing number of a graph is the smallest cardinality of a fixingý ýsetý. ýThe fixed number of a graph G is the minimum ký, ýsuch that ýevery k-set of vertices of G is a fixing set of Gý. ýA graph Gý ýis called a k-fixed graphý, ýif its fixing number and fixed numberý ýare both ký. ýIn this paperý, ýwe study the fixed number of a graphý ýand give a construction of a graph of higher fixed number from aý ýgraph of lower fixed numberý. ýWe find the bound on k in terms ofý ýthe diameter d of a distance-transitive k-fixed graphý.