Partial domination in prisms of graphs

For any graph G = (V, E) and proportion p E (0, 1], a set S C_ V is a p-dominating set if |N[S]| |V | > p. The p-domination number gamma p(G) equals the minimum cardinality of a p-dominating set in G. For a permutation pi of the vertex set of G, the graph pi G is obtained from two disjoint copies G1 and G2 of G by joining each v in G1 to pi(v) in G2. i.e., V (pi G) = V(G1) U V (G2) and E(G) = E(G1) U E(G2) U {(v, pi(v)) : v E V(G1), pi(v) E V(G2)}. The graph pi G is called the prism of G with respect to pi. In this paper, we find some relations between the domination and the p-domination numbers in the context of graph and its prism graph for particular values of p.