Finding odd cycle transversals

doi:10.1016/j.orl.2003.10.009

Operations Research Letters

Volume 32, Issue 4, July 2004, Pages 299-301

https://doi.org/10.1016/j.orl.2003.10.009 Get rights and content

Abstract

We present an O(mn) algorithm to determine whether a graph G with m edges and n vertices has an odd cycle transversal of order at most k, for any fixed k. We also obtain an algorithm that determines, in the same time, whether a graph has a half integral packing of odd cycles of weight k.

Section snippets

The algorithm

Finding a minimum transversal (or cover) for the set of odd cycles in a graph G is a fundamental problem in combinatorial optimization. Determining a minimum edge transversal is equivalent to the classic NP-complete problem, maximum cut [1]. Moreover, an easy reduction shows that the minimum vertex transversal problem contains the maximum cut problem as a special case. The vertex version also has direct applications in combinatorial biology (see, for example, [3]). In these applications, the

Transversals and flows

Given a graph G=(V,E), an odd cycle transversal X of G, and a partition of G−X into two stable sets S₁ and S₂, we construct an auxiliary bipartite graph G^′ as follows. The vertex set of the auxiliary graph is $V′=V−X+{x_{1},x_{2} : x∈X}$ . We maintain a one-to-one correspondence between the edges of G and the edges of the auxiliary graph by the following scheme:

(i)
For an edge e of G−X there is a corresponding edge in G^′ with the same endpoints.
(ii)
For an edge e∈G joining a vertex y in S_i to a vertex x in X the

Concluding remarks

It is shown in [2] that, for all k, there is an n_k such that every graph either has a set of 2k odd cycles such that each vertex is in at most two of these cycles, or it has an odd cycle transversal with at most n_k vertices. Furthermore, as described in the last section of [2], given an odd cycle transversal of order n_k, we can test for the existence of such a set of 2k odd cycles in O(mn) time. Using this and ODDCYCLETRANSVERSAL(n_k) we obtain the following result.

Theorem 3

For any fixed k, there is an O(

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R. Karp
Reducibility among combinatorial problems

There are more references available in the full text version of this article.

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