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The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover
| Citations: | 6 - 2 self |
BibTeX
@MISC{Mishra_thecomplexity,
author = {Sounaka Mishra and Venkatesh Raman and Saket Saurabh and Somnath Sikdar and C. R. Subramanian},
title = {The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover},
year = {}
}
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Abstract
A graph is König-Egerváry if the size of a minimum vertex cover equals that of a maximum matching in the graph. These graphs have been studied extensively from a graph-theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding König-Egerváry subgraphs of a given graph. In particular, given a graph G and a nonnegative integer k, we are interested in the following questions: 1. does there exist a set of k vertices (edges) whose deletion makes the graph König-Egerváry? 2. does there exist a set of k vertices (edges) that induce a König-Egerváry subgraph? We show that these problems are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. Towards this end, we first study the algorithmic complexity of Above Guarantee Vertex Cover, where one is interested in minimizing the additional number of vertices needed beyond the maximum matching size for the vertex cover. Further, while studying the parameterized complexity of the problem of deleting k vertices to obtain a König-Egerváry graph, we show a number of interesting structural results on matchings and vertex covers which could be useful in other contexts.
Keyphrases
nig subgraph problem above-guarantee vertex cover parameterized complexity algorithmic complexity vertex cover graph nig-egerv ry additional number nig-egerv ry graph nonnegative integer interesting structural result nig-egerv ry subgraphs minimum vertex cover equal maximum matching nig-egerv ry nig-egerv ry subgraph following question graph-theoretic point guarantee vertex cover maximum matching size
