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- Jason T. Hedetniemi, Kevin D. Hedetniemi, Sandra M. Hedetniemi, Stephen T. Hedetniemi
- 2009

For any given type of a set of vertices in a connected graph G = (V,E), we seek to determine the smallest integers (x, y : z) such that all minimal (or maximal) sets S of the given type, where |V | > |S| ≥ 2, have the property that every vertex v ∈ V − S is within distance at most x to a vertex u ∈ S (shortest distance), and within distance at most y to a… (More)

- Jason Hedetniemi
- J. Comb. Optim.
- 2016

- Jason Hedetniemi
- Discussiones Mathematicae Graph Theory
- 2015

Unique minimum vertex dominating sets in the Cartesian product of a graph with a complete graph are considered. We first give properties of such sets when they exist. We then show that when the first factor of the product is a tree, consideration of the tree alone is sufficient to determine if the product has a unique minimum dominating set.

- Jason Hedetniemi
- Australasian J. Combinatorics
- 2015

Unique minimum dominating sets in the Cartesian product of a graph and a Hamming graph are considered. A characterization of such sets is given, when they exist. A necessary and sufficient condition for the existence of a unique minimum dominating set is given in the special case of the Cartesian product of a tree and multiple copies of the same complete… (More)

- Jason Hedetniemi
- Australasian J. Combinatorics
- 2017

In this paper, we consider graphs having a unique minimum independent dominating set. We first discuss the effects of deleting a vertex, or the closed neighborhood of a vertex, from such graphs. We then discuss five operations which, in certain circumstances, can be used to combine two graphs, each having a unique minimum independent dominating set, to… (More)

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