Vertex domination of generalized Petersen graphs

Abstract

In a graph G a vertex v dominates all its neighbors and itself. A set D of vertices of G is (vertex) dominating set if each vertex of G is dominated by at least one vertex in D . The (vertex) domination number of G , denoted by γ ( G ) , is the cardinality of a minimum dominating set of G . A set D of vertices in G is efficient dominating set if every vertex of G is dominated by exactly one vertex of D . For natural numbers n and k , where n > 2 k , a generalized Petersen graph P ( n , k ) is obtained by letting its vertex set be { u 1 , u 2 , … , u n } ∪ { v 1 , v 2 , … , v n } and its edge set be the union of { u i u i + 1 , u i v i , v i v i + l } over 1 ≤ i ≤ n , where subscripts are reduced modulo n . We prove a necessary and sufficient condition for these graphs to have an efficient dominating set, and we determine exact values of γ ( P ( n , k ) ) for k ∈ { 1 , 2 , 3 } . Also we prove that for an odd number k , γ ( P ( n , k ) ) = n 2 + O ( k ) and for an even number k > 2 , γ ( P ( n , k ) ) ≤ 5 n 9 + O ( k ) .