A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. We show that for trees, the paired-domination number is less than or equal to the double domination number, solving a conjecture of Chellali and Haynes. Then we characterize the trees having equal paired and double domination numbers.

A paired dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. First a necessary and sufficient condition is given for a double dominating set (respectively, paired dominating set) to be minimal in G. We show that for clawfree graphs, the paired domination number is less than or equal to the double domination number. Then bounds on the double and paired domination numbers are presented. Sums involving these parameters are also considered.

In a graph G = (V,E), a subset S ⊆ V is a double dominating set if every vertex in V is dominated at least twice. The minimum cardinality of a double dominating set of G is the double domination number. A graph G is double domination edge critical if for any edge uv ε E(Ḡ), the double domination number of G + uv is less than the double domination number of G. We investigate double domination edge critical graphs and characterize the trees and cycles having this property. Then we concentrate on double domination edge critical graphs having small double domination numbers. In particular, we characterize the ones with double domination number three and subfamilies of those with double domination number four.

In a graph G=(V,E), a subset S ⊆ V is a double dominating set if every vertex in V is dominated at least twice. The minimum cardinality of a double dominating set of G is the double domination number. A graph G is double domination edge critical if for any edge uv ∈ E(G̅), the double domination number of G+uv is less than the double domination number of G. We investigate properties of double domination edge critical graphs. In particular, we characterize the double domination edge critical trees and cycles, graphs with double domination numbers of 3, and graphs with double domination numbers of 4 with maximum diameter.

The complementary prism GḠ of a graph G is formed from the disjoint union of G and its complement Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A set S ⊆ V(G) is a double dominating set of G if for every v ∈ V(G)\S, v is adjacent to at least two vertices of S, and for every w ∈ S, w is adjacent to at least one vertex of S. The double domination number of G is the minimum cardinality of a double dominating set of G. We begin by determining the double domination number of complementary prisms of paths and cycles. Then we characterize the graphs G whose complementary prisms have small double domination numbers. Finally, we establish lower and upper bounds on the double domination number of GḠ and show that all values between these bounds are attainable.

The complementary prism GḠ of a graph G is formed from the disjoint union of G and its complement Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A set S ⊆ V(G) is a double dominating set of G if for every v ∈ V(G)\S, v is adjacent to at least two vertices of S, and for every w ∈ S, w is adjacent to at least one vertex of S. The double domination number of G is the minimum cardinality of a double dominating set of G. We begin by determining the double domination number of complementary prisms of paths and cycles. Then we characterize the graphs G whose complementary prisms have small double domination numbers. Finally, we establish lower and upper bounds on the double domination number of GḠ and show that all values between these bounds are attainable.

The complementary prism of a graph G is obtained from a copy of G and its complement G̅ by adding a perfect matching between the corresponding vertices of G and G̅. For any graph G, a set D ⊆ V (G) is a double dominating set (DDS) if that set dominates every vertex of G twice. The double domination number, denoted γ×2(G), is the cardinality of a minimum double dominating set of G. We have proven results on graphs of small order, specific families and lower bounds on γ×2(GG̅).

We determine upper bounds on the ratios of several domination parameters in trees.