Total domination stable graphs upon edge addition

AbstractA set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination edge addition stable if the addition of an arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge addition stable graphs. We determine a sharp upper bound on the total domination number of total domination edge addition stable graphs, and we determine which combinations of order and total domination number are attainable. We finish this work with an investigation of claw-free total domination edge addition stable graphs

Restrained Domination in Complementary Prisms

The complementary prism GḠ of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A set S ⊆ V(G) is a restrained dominating set of G if for every v € V(G) \S, v is adjacent to a vertex in S and a vertex in V(G) \S. The restrained domination number of G is the minimum cardinality of a restrained dominating set of G. We study restrained domination of complementary prisms. In particular, we establish lower and upper bounds on the restrained domination number of GḠ, show that the restrained domination number can be attained for all values between these bounds, and characterize the graphs which attain the lower bound

Double Domination in Complementary Prisms

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

Partitioning the Vertices of a Graph into Two Total Dominating Sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition

Partitioning the vertices of a graph into two total dominating sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every self-complementary graph with minimum degree at least two has such a partition.Mathematics Subject Classification (2010): 05C69.Keywords: Total domination, vertex partitions, dominating sets, self-complementary graph

Roman Domination in Complementary Prisms

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect matching between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V, E) is a labeling f: V (G) → {0, 1, 2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) of G is the minimum f(V) = Σv∈V f(v) over all such functions of G. We study the Roman domination number of complementary prisms. Our main results show that γR(GG) takes on a limited number of values in terms of the domination number of GG and the Roman domination numbers of G and G

Double Domination in Complementary Prisms

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

Partitioning the Vertices of a Cubic Graph Into Two Total Dominating Sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study cubic graphs whose vertex set can be partitioned into two total dominating sets. There are infinitely many examples of connected cubic graphs that do not have such a vertex partition. In this paper, we show that the class of claw-free cubic graphs has such a partition. For an integer k at least 3, a graph is k-chordal if it does not have an induced cycle of length more than k. Chordal graphs coincide with 3-chordal graphs. We observe that for k≥6, not every graph in the class of k-chordal, connected, cubic graphs has two vertex disjoint total dominating sets. We prove that the vertex set of every 5-chordal, connected, cubic graph can be partitioned into two total dominating sets. As a consequence of this result, we observe that this property also holds for a connected, cubic graph that is chordal or 4-chordal. We also prove that cubic graphs containing a diamond as a subgraph can be partitioned into two total dominating sets

Domination Edge Lift Critical Trees

stract. Let uxv be an induced path with center x in a graph G. The edge lifting of uv off x is defined as the action of removing edges ux and vx from the edge set of G, while adding the edge uv to the edge set of G. We study trees for which every possible edge lift changes the domination number. We show that there are no trees for which every possible edge lift decreases the domination number. Trees for which every possible edge lift increases the domination number are characterized.Quaestiones Mathematicae 35(2012), 57–68