43 citations · 93 across the 31 of their papers we have counts for
41 papers
On the isolation numbers in graph products
Boštjan Brešar, Douglas F. Rall
As a continuation of a previous study of isolation numbers in Cartesian and lexicographic products, we investigate isolation numbers and, more generally, -isolation numbers…
Total isolation game in graphs
Michael A. Henning, Douglas F. Rall
The total isolation game is played on a graph by two players who take turns playing a vertex such that if is the set of already played vertices, then a vertex can be select…
On maximizing private neighbors in graphs
Stephen T. Hedetniemi, Douglas F. Rall
Given a set of vertices in a graph , a {\it private neighbor with respect to the set } is any vertex having precisely one neighbor, say ,…
Isolation number: Cartesian and lexicographic products and generalized Sierpiński graphs
Bostjan Bresar, Tanja Dravec, Daniel P. Johnston +3
The isolation number of a graph is the minimum cardinality of a set such that the subgraph induced by the vertices that are not in the union of the close…
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…
Claw-free cubic graphs are -colorable
Boštjan Brešar, Kirsti Kuenzel, Douglas F. Rall
A -coloring of a graph is a partition of its vertex set into four sets two of which are independent and the other two are -packings. In this paper, we prove that ever…
Isolation game on graphs
Boštjan Brešar, Tanja Dravec, Daniel P. Johnston +2
Given a graph and a family of graphs , an -isolating set, as introduced by Caro and Hansberg, is any set such that contains no member…
Indicated domination game
Boštjan Brešar, Csilla Bujtás, Vesna Iršič +2
Motivated by the success of domination games and by a variation of the coloring game called the indicated coloring game, we introduce a version of domination games called the indic…
Indicated total domination game
Michael A. Henning, Douglas F. Rall
A vertex in a graph totally dominates a vertex if is adjacent to in . A total dominating set of is a set of vertices of such that every vertex of…
Packings in bipartite prisms and hypercubes
Boštjan Brešar, Sandi Klavžar, Douglas F. Rall
The -packing number of a graph is the cardinality of a largest -packing of and the open packing number is the cardinality of a largest open pa…
A characterization of well-dominated Cartesian products
Kirsti Kuenzel, Douglas F. Rall
A graph is well-dominated if all its minimal dominating sets have the same cardinality. In this paper we prove that at least one factor of every connected, well-dominated Cartesian…
Graphs with equal Grundy domination and independence number
Gábor Bacsó, Boštjan Brešar, Kirsti Kuenzel +1
The Grundy domination number, , of a graph is the maximum length of a sequence of vertices in such that for every $i\in \{2,\ldots,…
Orientable domination in product-like graphs
Sarah Anderson, Boštjan Brešar, Sandi Klavžar +2
The orientable domination number, , of a graph is the largest domination number over all orientations of . In this paper, is studied on different p…
On independent domination in direct products
Kirsti Kuenzel, Douglas F. Rall
In \cite{nr-1996} Nowakowski and Rall listed a series of conjectures involving several different graph products. In particular, they conjectured that wh…
On well-edge-dominated graphs
Sarah E. Anderson, Kirsti Kuenzel, Douglas F. Rall
A graph is said to be well-edge-dominated if all its minimal edge dominating sets are minimum. It is known that every well-edge-dominated graph is also equimatchable, meaning t…
On well-dominated direct, Cartesian and strong product graphs
Douglas F. Rall
If each minimal dominating set in a graph is a minimum dominating set, then the graph is called well-dominated. Since the seminal paper on well-dominated graphs appeared in 1988, t…
Domination in digraphs and their products
Boštjan Brešar, Kirsti Kuenzel, Douglas F. Rall
A dominating (respectively, total dominating) set of a digraph is a set of vertices in such that the union of the closed (respectively, open) out-neighborhoods of verti…
The Enclaveless Competition Game
Michael A. Henning, Douglas F. Rall
For a subset of vertices in a graph , a vertex is an enclave of if and all of its neighbors are in , where a neighbor of is a vertex adjacent to …
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…
General -position sets
Sandi Klavzar, Douglas F. Rall, Ismael G. Yero
The general -position number of a graph is the cardinality of a largest set for which no three distinct vertices from lie on a common geodesic of len…
On well-dominated graphs
Sarah E. Anderson, Kirsti Kuenzel, Douglas F. Rall
A graph is \emph{well-dominated} if all of its minimal dominating sets have the same cardinality. We prove that at least one of the factors is well-dominated if the Cartesian produ…
Graphs with a unique maximum open packing
Boštjan Brešar, Kirsti Kuenzel, Douglas F. Rall
A set of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in are disjoint. In this paper, we consider the graphs that have a uniq…
On well-covered direct products
Kirsti Kuenzel, Douglas F. Rall
A graph is well-covered if all maximal independent sets of have the same cardinality. In 1992 Topp and Volkmann investigated the structure of well-covered graphs that have…
Packing chromatic vertex-critical graphs
Sandi Klavžar, Douglas F. Rall
The packing chromatic number of a graph is the smallest integer such that the vertex set of can be partitioned into sets , , where vertices in $…