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Connected domination in grid graphs.
Masahisa Goto,Koji M. Kobayashi +1 more
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In this article, the minimum cardinality of a connected dominating set, called the connected domination number (CDP), of an undirected simple graph was determined for any $m \times n$ grid graph.Abstract:
Given an undirected simple graph, a subset of the vertices of the graph is a {\em dominating set} if every vertex not in the subset is adjacent to at least one vertex in the subset. A subset of the vertices of the graph is a {\em connected dominating set} if the subset is a dominating set and the subgraph induced by the subset is connected. In this paper, we determine the minimum cardinality of a connected dominating set, called the {\em connected domination number}, of an $m \times n$ grid graph for any $m$ and $n$.read more
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Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Journal ArticleDOI
Unit disk graphs
Brent N. Clark,Brent N. Clark,Charles J. Colbourn,Charles J. Colbourn,David S. Johnson,David S. Johnson +5 more
TL;DR: It is shown that many standard graph theoretic problems remain NP-complete on unit disks, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs.
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Approximation Algorithms for Connected Dominating Sets
TL;DR: The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex.
Journal ArticleDOI
Hamilton Paths in Grid Graphs
TL;DR: This work gives necessary and sufficient conditions for the graph to have a Hamilton path between these two nodes, and provides a new, relatively simple, proof of the result that the Euclidean traveling salesman problem is NP-complete.
Journal ArticleDOI
Clustering and domination in perfect graphs
TL;DR: This paper investigates the complexity status of these problems on various sub-classes of perfect graphs, including comparability graphs, chordal graphs, bipartite graphs, split graphs, cographs and κ-trees, where the k-cluster problem is polynomial and the weighted and connected versions are studied.