Rodulfo T. Aunzo, Enrico L. Enriquez
Let G be a connected simple graph. A set S ⊆ V (G) is a doubly connected dominating set if it is dominating and both 〈 S〉 and 〈 V (G)\S 〉 are connected. The doubly connected domination number of G, denoted by γcc(G), is the smallest cardinality of a doubly connected dominating set S of G. A convex dominating set S of G is a convex doubly connected dominating set if S is a doubly connected dominating set of G. The convex doubly connected domination number of G, denoted by γccc(G), is the smallest cardinality of a convex doubly connected dominating set S of G. In this paper, we show that every integers a, b, c, and n with 1 ≤ a ≤ b ≤ c ≤ n is realizable as domination number, convex domination number, convex doubly connected domination number, and order of G respectively. Further, we give the characterization of the convex doubly connected dominating set with convex doubly connected domination numbers of 1 and 2. Finally, we characterize the convex doubly connected dominating sets of the join and corona of graphs. © 2015 Rodulfo T. Aunzo, Jr. and Enrico L. Enriquez.
Department of Mathematics, School of Arts and Sciences, University of San Carlos, Cebu City, 6000, Philippines
