5 papers
The domatic number game played on graphs
Bert L. Hartnell, Douglas F. Rall
The domatic number of a graph is the maximum number of pairwise disjoint dominating sets admitted by the graph. We introduce a game based around this graph invariant. The domatic n…
On graphs having one size of maximal open packings
Bert L. Hartnell, Douglas F. Rall
A set of vertices in a graph is an open packing if no two distinct vertices in have a common neighbor. Among all maximal open packings in , the smallest cardinality…
On well-covered Cartesian products
Bert L. Hartnell, Douglas F. Rall, Kirsti Wash
In 1970, Plummer defined a well-covered graph to be a graph in which all maximal independent sets are in fact maximum. Later Hartnell and Rall showed that if the Cartesian prod…
On the Cartesian product of non well-covered graphs
Bert L. Hartnell, Douglas F. Rall
A graph is well-covered if every maximal independent set has the same cardinality, namely the vertex independence number. We answer a question of Topp and Volkmann and prove that i…
On graphs having maximal independent sets of exactly distinct cardinalities
Bert L. Hartnell, Douglas F. Rall
For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of…