Complexity of Secure Sets

A secure set S in a graph is defined as a set of vertices such that for any X ⊆ S the majority of vertices in the neighborhood of X belongs to S . It is known that deciding whether a set S is secure in a graph is co- NP -complete. However, it is still open how this result contributes to the actual c...

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Bibliographic Details
Published in:Algorithmica Vol. 80; no. 10; pp. 2909 - 2940
Main Authors: Bliem, Bernhard, Woltran, Stefan
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2018
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Mathematics of Computing
Theory of Computation
Upper bounds
Complexity analysis
Parameterized algorithms
Secure set
Parameterized complexity
Treewidth
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:A secure set S in a graph is defined as a set of vertices such that for any X ⊆ S the majority of vertices in the neighborhood of X belongs to S . It is known that deciding whether a set S is secure in a graph is co- NP -complete. However, it is still open how this result contributes to the actual complexity of deciding whether for a given graph G and integer k , a non-empty secure set for G of size at most k exists. In this work, we pinpoint the complexity of this problem by showing that it is Σ 2 P -complete. Furthermore, the problem has so far not been subject to a parameterized complexity analysis that considers structural parameters. In the present work, we prove that the problem is W [ 1 ] -hard when parameterized by treewidth. This is surprising since the problem is known to be FPT when parameterized by solution size and “subset problems” that satisfy this property usually tend to be FPT for bounded treewidth as well. Finally, we give an upper bound by showing membership in XP , and we provide a positive result in the form of an FPT algorithm for checking whether a given set is secure on graphs of bounded treewidth.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-017-0358-5