Displaying similar documents to “On Unique Minimum Dominating Sets in Some Cartesian Product Graphs”

On Graphs with Disjoint Dominating and 2-Dominating Sets

Michael A. Henning, Douglas F. Rall (2013)

Discussiones Mathematicae Graph Theory

Similarity:

A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only connected graph with minimum degree at least three for which D ∪ D2 necessarily contains all vertices...

Trees with equal 2-domination and 2-independence numbers

Mustapha Chellali, Nacéra Meddah (2012)

Discussiones Mathematicae Graph Theory

Similarity:

Let G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V-S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number γ₂(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β₂(G). Fink and Jacobson proved that γ₂(G) ≤ β₂(G) for every graph G. In this paper we provide a constructive characterization...

On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

Fatemeh Alinaghipour Taklimi, Shaun Fallat, Karen Meagher (2014)

Special Matrices

Similarity:

The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing...

Minimum vertex ranking spanning tree problem for chordal and proper interval graphs

Dariusz Dereniowski (2009)

Discussiones Mathematicae Graph Theory

Similarity:

A vertex k-ranking of a simple graph is a coloring of its vertices with k colors in such a way that each path connecting two vertices of the same color contains a vertex with a bigger color. Consider the minimum vertex ranking spanning tree (MVRST) problem where the goal is to find a spanning tree of a given graph G which has a vertex ranking using the minimal number of colors over vertex rankings of all spanning trees of G. K. Miyata et al. proved in [NP-hardness proof and an approximation...

Trees with unique minimum total dominating sets

Teresa W. Haynes, Michael A. Henning (2002)

Discussiones Mathematicae Graph Theory

Similarity:

A set S of vertices of a graph G is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. We provide three equivalent conditions for a tree to have a unique minimum total dominating set and give a constructive characterization of such trees.