# Feedback Vertex Sets on Tree Convex Bipartite Graphs

• Chaoyi Wang
• Tian Liu
• Wei Jiang
• Ke Xu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7402)

## Abstract

A feedback vertex set in a graph is a subset of vertices, such that the complement of this subset induces a forest. Finding a minimum feedback vertex set (FVS) is $$\cal{NP}$$-complete on bipartite graphs, but tractable on chordal bipartite graphs. A bipartite graph is called tree convex, if a tree is defined on one part of the vertices, such that for every vertex in the other part, the neighborhood of this vertex induces a subtree. First, we show that chordal bipartite graphs form a proper subset of tree convex bipartite graphs. Second, we show that FVS is $$\cal{NP}$$-complete on the tree convex bipartite graphs where the sum of the degrees of vertices whose degree is at least three on the tree is unbounded. Combined with known tractability where this sum is bounded, we show a dichotomy of complexity of FVS on tree convex bipartite graphs.

## Keywords

Bipartite Graph Common Element Vertex Cover Fourth Stage Complete Bipartite Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Chaoyi Wang
• 1
• Tian Liu
• 1
• Wei Jiang
• 1
• Ke Xu
• 2
1. 1.Key Laboratory of High Confidence Software Technologies, Ministry of Education, Institute of Software, School of Electronic Engineering and Computer SciencePeking UniversityBeijingChina
2. 2.National Lab of Software Development EnvironmentBeihang UniversityBeijingChina