8,834 research outputs found
Minimal Restrained Domination Algorithms on Trees Using Dynamic Programming
In this paper we study a special case of graph domination, namely minimal restrained dominating sets on trees. A set S ? V is a dominating set if for every vertex u ? V-S, there exists v ? S such that uv ? E. A set S ? V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and another vertex in V-S. A restrained dominating set S is a minimal restrained dominating set if no proper subset of S is also a restrained dominating set. We give a dynamic programming style algorithm for generating largest minimal restrained dominating sets for trees and show that the decision problem for minimal restrained dominating sets is NP-complete for general graphs. We also consider independent restrained domination on trees and its associated decision problem for general graphs
On Roman, Global and Restrained Domination in Graphs
In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination number, and the total restrained domination number is equal to the total domination number. A number of open problems are posed. © 2010 Springer
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
Bipartitions Based on Degree Constraints
For a graph G = (V,E), we consider a bipartition {V1,V2} of the vertex set V by placing constraints on the vertices as follows. For every vertex v in Vi, we place a constraint on the number of neighbors v has in Vi and a constraint on the number of neighbors it has in V3-i. Using three values, namely 0 (no neighbors are allowed), 1 (at least one neighbor is required), and X (any number of neighbors are allowed) for each of the four constraints, results in 27 distinct types of bipartitions. The goal is to characterize graphs having each of these 27 types. We give characterizations for 21 out of the 27. Three other characterizations appear in the literature. The remaining three prove to be quite difficult. For these, we develop properties and give characterization of special families
Trees whose 2-domination subdivision number is 2
A set of vertices in a graph is a -dominating set if every vertex of is adjacent to at least two vertices of . The -domination number of a graph , denoted by , is the minimum size of a -dominating set of . The -domination subdivision number is the minimum number of edges that must be subdivided (each edge in can be subdivided at most once) in order to increase the -domination number. The authors have recently proved that for any tree of order at least , . In this paper we provide a constructive characterization of the trees whose -domination subdivision number is
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