A complexity dichotomy and a new boundary class for the dominating set problem

We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the compl...

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Bibliographic Details
Published in:Journal of combinatorial optimization Vol. 32; no. 1; pp. 226 - 243
Main Author: Malyshev, D. S.
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2016
Subjects:
Combinatorics
Convex and Discrete Geometry
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Theory of Computation
Boundary class
Polynomial-time algorithm
Dominating set
Computational complexity
ISSN:1382-6905, 1573-2886
Online Access:Get full text
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Summary:We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the complexity of the problem even for hereditary classes with small forbidden structures. We completely determine the complexity of the problem for classes defined by forbidding a five-vertex path and any set of fragments with at most five vertices. Additionally, we also prove polynomial-time solvability of the problem for some two classes of a similar type. The notion of a boundary class is a helpful tool for analyzing the computational complexity of graph problems in the family of hereditary classes. Three boundary classes were known for the dominating set problem prior to this paper. We present a new boundary class for it.
ISSN:1382-6905
1573-2886
DOI:10.1007/s10878-015-9872-z