47 research outputs found
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Eternal Independent Sets in Graphs
The use of mobile guards to protect a graph has received much attention in the literature of late in the form of eternal dominating sets, eternal vertex covers and other models of graph protection. In this paper, eternal independent sets are introduced. These are independent sets such that the following can be iterated forever: a vertex in the independent set can be replaced with a neighboring vertex and the resulting set is independent
Disjoint Dominating Sets with a Perfect Matching
In this paper, we consider dominating sets and such that and
are disjoint and there exists a perfect matching between them. Let
denote the cardinality of smallest such sets in
(provided they exist, otherwise ). This
concept was introduced in [Klostermeyer et al., Theory and Application of
Graphs, 2017] in the context of studying a certain graph protection problem. We
characterize the trees for which equals a certain
graph protection parameter and for which ,
where is the independence number of . We also further study this
parameter in graph products, e.g., by giving bounds for grid graphs, and in
graphs of small independence number
Vertex covers and eternal dominating sets
AbstractThe eternal domination problem requires a graph to be protected against an infinitely long sequence of attacks on vertices by guards located at vertices, the configuration of guards inducing a dominating set at all times. An attack at a vertex with no guard is defended by sending a guard from a neighboring vertex to the attacked vertex. We allow any number of guards to move to neighboring vertices at the same time in response to an attack. We compare the eternal domination number with the vertex cover number of a graph. One of our main results is that the eternal domination number is less than the vertex cover number of any graph of minimum degree at least two having girth at least nine
Edge Dominating Sets and Vertex Covers
Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of trees and grid graphs
A Taxonomy of Perfect Domination
Abstract: A variety of terminology has been used in the literature to describe a dominating set with the property that each vertex in graph is dominated exactly once (or at most once, depending on the situation). We review the various terminologies related to perfect domination, consider some of the variations of perfect domination studied in the literature, and survey some of the main results
A Dynamic Domination Problem
A dynamic domination problem in graphs is studied in which an innite sequence of attacks occurs at vertices with guards and the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex. Other guards are allowed to move at the same time, and before and after each attack, the vertices containing guards must form a dominating set. We are concerned with the minimum number of guards required to achieve this goal against any arbitrary sequence of attacks; this number is called the swap number. The swap number lies between the domination and independence numbers of the graph; bounds for some classes of graphs are examined in this paper