B. Chaluvaraju

Generalized perfect domination in graphs

Let k be a positive integer and G=(V,E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G, if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γkp(G). In this paper, we give characterizations of graphs for which γkp(G)=γ(G)+k−2 and prove that the perfect k-domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs. Also, by using dynamic programming techniques, we obtain an algorithm to determine the perfect k-domination number of trees.

On coefficients of edge domination polynomial of a graph

Abstract An edge domination polynomial of a graph G is the polynomial where de(G, t) is the number of edge dominating sets of G of cardinality t. In this paper, we provide tables which contain coefficient of edge domination polynomial of path and cycle. Also, certain properties of edge dominating polynomial are given.

Complementary total domination in graphs

Abstract Let D be a minimum total dominating set of G. If V−D contains a total dominating set (TDS) say S of G, then S is called a complementary total dominating set with respect to D. The complementary total domination number γ ct (G) of G is the minimum number of vertices in a complementary total dominating set (CTDS) of G .In this paper, exact values of γ ct (G) for some standard graphs are obtained. Also its relationship with other domination related parameters are investigated.