BOUNDS ON THE DOMINATION NUMBER OF PERMUTATION GRAPHS

For a graph G with n vertices and a permutation alpha on V (G), a permutation graph P-alpha(G) is obtained from two identical copies of G by adding an edge between v and alpha(v) for any v is an element of V (G). Let gamma(G) be the domination number of a graph G. It has been shown that gamma(G) <= gamma (P-alpha (G)) <= 2 gamma(G) for any permutation alpha on V (G). In this paper, we investigate specific graphs for which there exists a permutation alpha such that gamma(P-alpha (G)) > gamma(G) in terms of the domination number of G or the maximum degree of G. Additionally, we construct a class of graphs for which the domination number of any permutation graph is twice the domination number of the original graph, as well as explore finding a specific graph G and permutation alpha for any two positive integers a and b with a <= b <= 2a, to have gamma(G) = a and gamma(P-alpha(G)) = b.