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Teresa W. Haynes and Michael A. Henning

Abstract

Let G be a graph with vertex set V and no isolated vertices. A sub-set SV is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number γpr2(G) is the minimum cardinality of a semipaired dominating set of G. We show that if G is a connected graph G of order n ≥ 3, then γpr2(G)23n, and we characterize the extremal graphs achieving equality in the bound.

Open access

Wyatt J. Desormeaux, Teresa W. Haynes and Michael A. Henning

Abstract

A dominating set in a graph G is a set S of vertices such that every vertex in V (G) \ S is adjacent to at least one vertex in S, and the domination number of G is the minimum cardinality of a dominating set of G. Placing constraints on a dominating set yields different domination parameters, including total, connected, restrained, and clique domination numbers. In this paper, we study relationships among domination parameters of a graph and its complement.

Open access

Wyatt J. Desormeaux, Teresa W. Haynes and Michael A. Henning

Abstract

A set S of vertices of a graph G is a dominating set if every vertex not in S is adjacent to a vertex of S and is a total dominating set if every vertex of G is adjacent to a vertex of S. The cardinality of a minimum dominating (total dominating) set of G is called the domination (total domination) number. A set that does not dominate (totally dominate) G is called a non-dominating (non-total dominating) set of G. A partition of the vertices of G into non-dominating (non-total dominating) sets is a non-dominating (non-total dominating) set partition. We show that the minimum number of sets in a non-dominating set partition of a graph G equals the total domination number of its complement G̅ and the minimum number of sets in a non-total dominating set partition of G equals the domination number of G̅ . This perspective yields new upper bounds on the domination and total domination numbers. We motivate the study of these concepts with a social network application.

Open access

Robert A. Beeler, Teresa W. Haynes and Kyle Murphy

Abstract

Let G be a graph with vertex set V and a distribution of pebbles on the vertices of V. A pebbling move consists of removing two pebbles from a vertex and placing one pebble on a neighboring vertex, and a rubbling move consists of removing a pebble from each of two neighbors of a vertex v and placing a pebble on v. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than one pebble and for any given vertex vV, it is possible, by a sequence of pebbling and rubbling moves, to move at least one pebble to v. This minimum number of pebbles is the 1-restricted optimal rubbling number. We determine the 1-restricted optimal rubbling numbers for Cartesian products. We also present bounds on the 1-restricted optimal rubbling number.

Open access

Teresa W. Haynes, Stephen T. Hedetniemi, Jessie D. Jamieson and William B. Jamieson

Abstract

A path π = (v1, v2, . . . , vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi) ≥ deg(vi+1), where deg(vi) denotes the degree of vertex vi ∈ V. The downhill domination number equals the minimum cardinality of a set S ⊆ V having the property that every vertex v ∈ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers of graphs and give upper bounds. In particular, we show that the downhill domination number of a graph is at most half its order, and that the downhill domination number of a tree is at most one third its order. We characterize the graphs obtaining each of these bounds

Open access

Gary Chartrand, Stephen Devereaux, Teresa W. Haynes, Stephen T. Hedetniemi and Ping Zhang

Abstract

Let G be a nontrivial connected, edge-colored graph. An edge-cut R of G is called a rainbow cut if no two edges in R are colored the same. An edge-coloring of G is a rainbow disconnection coloring if for every two distinct vertices u and v of G, there exists a rainbow cut in G, where u and v belong to different components of GR. We introduce and study the rainbow disconnection number rd(G) of G, which is defined as the minimum number of colors required of a rainbow disconnection coloring of G. It is shown that the rainbow disconnection number of a nontrivial connected graph G equals the maximum rainbow disconnection number among the blocks of G. It is also shown that for a nontrivial connected graph G of order n, rd(G) = n−1 if and only if G contains at least two vertices of degree n − 1. The rainbow disconnection numbers of all grids Pm _ Pn are determined. Furthermore, it is shown for integers k and n with 1 ≤ kn − 1 that the minimum size of a connected graph of order n having rainbow disconnection number k is n + k − 2. Other results and a conjecture are also presented.