On fixing sets of composition and corona product of graphs

A fixing set 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) circle dot 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 >= 1 components G(2)(1), G(2)(2), mn 1 - >= fix(G(1)[G(2)]) >= m fix(Sigma(i)(i=1) fix (G(2)(i))) For a connected graph G(1) and an arbitrary graph G(2) which are not asymmetric, we prove that fix(G(1)circle dot 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) circle dot G2) = max{f ix(G(1)), m f ix(G(2))}.