Corpus ID: 18916440

Partitioning Graphs into Generalized Dominating Sets

@article{Heggernes1998PartitioningGI,
  title={Partitioning Graphs into Generalized Dominating Sets},
  author={Pinar Heggernes and Jan Arne Telle},
  journal={Nord. J. Comput.},
  year={1998},
  volume={5},
  pages={128-142}
}
We study the computational complexity of partitioning the vertices of a graph into generalized dominating sets. Generalized dominating sets are parameterized by two sets of nonnegative integers σ and ρ which constrain the neighborhood N(υ) of vertices. A set S of vertices of a graph is said to be a (σ, ρ)-set if ∀υ ∈ S : |N(υ) ∩ S| ∈ σ and ∀υ n ∈ S : |N(υ) ∩ S| ∈ ρ. The (k, σ, ρ)-partition problem asks for the existence of a partition V1, V2, ..., Vk of vertices of a given graph G such that Vi… Expand
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References

SHOWING 1-10 OF 16 REFERENCES
Nearly perfect sets in graphs
TLDR
N p ( G) is defined to be the minimum cardinality of a 1-maximal nearly perfect set, and N p (G ) to beThe maximum cardinalityof a 1 -minimal nearlyperfect set. Expand
Vertex partitioning problems: characterization, complexity and algorithms on partial K-trees
This thesis investigates the computational complexity of algorithmic problems defined on graphs. At the abstract level of the complexity spectrum we discriminate polynomial-time solvable problemsExpand
Complexity of Domination-Type Problems in Graphs
  • J. A. Telle
  • Mathematics, Computer Science
  • Nord. J. Comput.
  • 1994
TLDR
This work gives a characterization of graph parameters that unifies their definitions, facilitates their common algorithmic treatment and allows for their uniform complexity classification, and provides the basis for a taxonomy of domination-type and independence-type problems. Expand
Algorithms for Vertex Partitioning Problems on Partial k-Trees
TLDR
A design methodology of practical solution algorithms for generally $\NP$-hard problems when restricted to partial k-trees (graphs with treewidth bounded by k) is presented, which accounts for dependency on the parameter k of the computational complexity of the resulting algorithms. Expand
A Best Possible Heuristic for the k-Center Problem
TLDR
A 2-approximation algorithm for the k-center problem with triangle inequality is presented, the key combinatorial object used is called a strong stable set, and the NP-completeness of the corresponding decision problem is proved. Expand
NP Completeness of Finding the Chromatic Index of Regular Graphs
Abstract We show that it is NP complete to determine whether it is possible to edge color a regular graph of degree k with k colors for any k ⩾ 3. As a by-product of this result, we obtain a new wayExpand
Bibliography on domination in graphs and some basic definitions of domination parameters
TLDR
The following bibliography on Domination in Graphs has been compiled over the past six years at Clemson University, where it is expected that this bibliography will continue to grow at a steady rate. Expand
The complexity of satisfiability problems
TLDR
An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete. Expand
Computers and Intractability: A Guide to the Theory of NP-Completeness
Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledgeExpand
The NP-Completeness of Edge-Coloring
  • I. Holyer
  • Mathematics, Computer Science
  • SIAM J. Comput.
  • 1981
TLDR
It is shown that it is NP-complete to determine the chromatic index of an arbitrary graph, even for cubic graphs. Expand
...
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