F of an undirected graph G is a vertex subset of G whose removal results in a forest. Such a set F is said to be independent if F forms an independent set of G. In this paper, we study the problem of finding an independent feedback vertex set of a given graph with the minimum number of vertices, from the viewpoint of graph classes. This problem is NP-hard even for planar bipartite graphs of maximum degree four. However, we show that the problem is solvable in linear time for graphs having tree-like structures, more specifically, for bounded treewidth graphs, chordal graphs and cographs. We then give a fixed-parameter algorithm for planar graphs when parameterized by the solution size. Such a fixed-parameter algorithm is already known for general graphs, but our algorithm is exponentially faster than the known one." />
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Algorithms for the Independent Feedback Vertex Set Problem

Yuma TAMURA
Takehiro ITO
Xiao ZHOU

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E98-A    No.6    pp.1179-1188
Publication Date: 2015/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E98.A.1179
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
Category: 
Keyword: 
fixed parameter tractability,  graph algorithm,  independent feedback vertex set,  

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Summary: 
A feedback vertex set F of an undirected graph G is a vertex subset of G whose removal results in a forest. Such a set F is said to be independent if F forms an independent set of G. In this paper, we study the problem of finding an independent feedback vertex set of a given graph with the minimum number of vertices, from the viewpoint of graph classes. This problem is NP-hard even for planar bipartite graphs of maximum degree four. However, we show that the problem is solvable in linear time for graphs having tree-like structures, more specifically, for bounded treewidth graphs, chordal graphs and cographs. We then give a fixed-parameter algorithm for planar graphs when parameterized by the solution size. Such a fixed-parameter algorithm is already known for general graphs, but our algorithm is exponentially faster than the known one.

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