On the odd-minor variant of Hadwiger's conjecture

Geelen, Jim; Gerards, Bert; Reed, Bruce; Seymour, Paul; Vetta, Adrian

doi:10.1016/j.jctb.2008.03.006

Cited by 62 publications

(66 citation statements)

References 11 publications

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“…So, we are left with the case that there is a huge flat proper subwall W of the wall W , which is bipartite. In this case, by the above result of Geelen et al [10], we can conclude either that the graph "attached" to a specific highly connected part of the graph W t is "essentially" bipartite in W t , (and hence in this case, this negates our need to construct the tree-decomposition, as discussed for the case 1), or there are many parity breaking paths, each joining two points on the outer cycle of the wall W in W t . By a parity breaking path, we mean a path P with endpoints on the outer cycle of the wall W such that the path P together with the wall W yields an odd cycle.…”

Section: A Wall Of Huge Height But No Huge Clique Minormentioning

confidence: 58%

“…It turns out that we have a nice structure theorem which tells us that if we have a huge clique minor, then either we can get a big odd clique minor or else we can get a vertex set X of bounded size (depending on k) such that the component of G − X containing most of the nodes of the huge clique minor is "essentially" bipartite. This is proved in Theorem 5.2 with help of the recent result by Geelen et al [10] (Actually, Theorem 5.2 essentially follows from the result in [10]. ).…”

Section: A Wall Of Huge Height But No Huge Clique Minormentioning

confidence: 81%

“…We shall use the following recent result in [10]. The proof given in [10] actually implies that there is a polynomial time algorithm for Theorem 4.1.…”

Section: Odd S-pathsmentioning

confidence: 99%

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An (almost) Linear Time Algorithm For Odd Cycles Transversal

Kawarabayashi¹,

Reed²

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the following problem, which is called the odd cycles transversal problem. Input: A graph G and an integer k.Output : A vertex set X ∈ V (G) with |X| ≤ k such that G − X is bipartite.We present an O(mα(m, n)) time algorithm for this problem for any fixed k, where n, m are the number of vertices and the number of edges, respectively, and the function α(m, n) is the inverse of the Ackermann function (see by Tarjan [38]).This improves the time complexity of the algorithm by Reed, Smith and Vetta [29] who gave an O(nm) time algorithm for this problem. Our algorithm also implies the edge version of the problem, i.e, there is an edge set X ∈ E(G) such that G − X is bipartite.Using this algorithm and the recent result in [16], we give an O(mα(m, n) + n log n) algorithm for the following problem for any fixed k:Input: A graph G and an integer k.Output : Determine whether or not there is a half-integral k disjoint odd cycles packing, i.e, k odd cycles C1, . . . , C k in G such that each vertex is on at most two of these odd cycles.This improves the time complexity of the algorithm by Reed, Smith and Vetta [29] who gave an O(n 3 ) time algorithm for this problem.We also give a much simpler and much shorter proof for the following result by Reed [28].The Erdős-Pósa property holds for the half-integral disjoint odd cycles packing problem. I.e. either G has a half-integral k * This work was done as a part of an INRIA-NII collaboration under MOU grant, and partially supported by MEXT Grant-in-Aid for Scientific Research on Priority Areas "New Horizons in Computing" † National Institute of Informatics, 2-1-2, Hitotsubashi, Chiyoda-ku, Tokyo, Japan. Email address: breed@cs.mcgill.ca disjoint odd cycles packing or G has a vertex set X of order at most f (k) such that G − X is bipartite for some function f of k.Note that the Erdős-Pósa property does not hold for odd cycles in general.

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Section: A Wall Of Huge Height But No Huge Clique Minormentioning

confidence: 58%

Section: A Wall Of Huge Height But No Huge Clique Minormentioning

confidence: 81%

See 1 more Smart Citation

An (almost) Linear Time Algorithm For Odd Cycles Transversal

Kawarabayashi¹,

Reed²

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Attempts to strengthen the 4CT so that it provides some bound in such cases has developed the theory of signed graphs. Coloring of graphs with signed graphs as forbidden minors have been studied, see for example Odd Hadwiger's conjecture (we refer to [8] for some recent developments), an extension of the well-known Hadwiger conjecture. Only recently, the development of the theory of homomorphisms of signed graphs has begun, see [10,14].…”

Section: Theorem 2 Deciding If a Given Planar Graph Is 3-colorable Imentioning

confidence: 99%

The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction

Foucaud

Naserasr

2014

LATIN 2014: Theoretical Informatics

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Abstract. A signed graph (G, Σ) is an undirected graph G together with an assignment of signs (positive or negative) to all its edges, where Σ denotes the set of negative edges. Two signatures are said to be equivalent if one can be obtained from the other by a sequence of resignings (i.e. switching the sign of all edges incident to a given vertex). Extending the notion of usual graph homomorphisms, homomorphisms of signed graphs were introduced, and have lead to some extensions and strengthenings in the theory of graph colorings and homomorphisms. We study the complexity of deciding whether a given signed graph admits a homomorphism to a fixed target signed graph [H, Σ], i.e. the (H, Σ)-Coloring problem. We prove a dichotomy result for the class of all (C k , Σ)-Coloring problems (where C k is a cycle of length k ≥ 3): (C k , Σ)-Coloring is NP-complete, unless both k and the size of Σ are even. We conjecture that this dichotomy can be extended to all signed graphs in a natural way. We also introduce the more general concept of signed constraint satisfaction problems and show that a dichotomy for such problems is equivalent to the statement of the Feder-Vardi Dichotomy Conjecture.

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“…structures under parity constraints; for example, having vertex-or edge-disjoint odd cycles [29,28], parity linkages [48,30,25], odd minors [19,31,33], odd subdivisions [27], vertex-disjoint packings of odd paths with ends in a specified vertex-set [4,3], and packing non-zero cycles in group-labelled graphs [22,32]. A key question for each type of structure is whether the Erdős-Pósa property holds: is there a function f such that there are either k vertex-disjoint (or edge-disjoint) instances of the structure or a set of f (k) vertices (or edges) intersecting all such instances?…”

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confidence: 99%

A submodular measure and approximate Gomory-Hu theorem for packing odd trails

Churchley

Mohar

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Motivated by a problem about totally odd immersions of graphs, we define the odd edge-connectivity λ o (u, v) as the maximum number of edge-disjoint trails of odd length from u to v. It was recently discovered that λ o (u, v) can be approximated up to a constant multiplicative factor using the usual edge-connectivity between u and v and the minimum value of another parameter that measures "how far from a bipartite graph" the part of the graph around u and v is.In this paper, we formalize this second ingredient and call it the perimeter. We prove that perimeter is a submodular function on the vertex-sets of a graph. Using this fact, we obtain a version of the Gomory-Hu Theorem in which minimum edge-cuts are replaced by sets of minimum perimeter. We construct (in polynomial time) a rooted forest structure, analogous to the Gomory-Hu tree of a graph, which encodes a collection of minimum-perimeter vertex-sets. Although the classical Gomory-Hu Theorem extends to arbitrary symmetric submodular functions, our result is novel and indicates a possibility for further generalizations.These results have significant implications for the study of path and trail systems with parity constraints. We present two such applications: an efficient data structure for storing approximate odd edge-connectivities for all pairs of vertices in a graph, and a rough structure theorem for graphs with no "totally odd" immersion of a large complete graph.

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On the odd-minor variant of Hadwiger's conjecture

Cited by 62 publications

References 11 publications

An (almost) Linear Time Algorithm For Odd Cycles Transversal

An (almost) Linear Time Algorithm For Odd Cycles Transversal

The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction

A submodular measure and approximate Gomory-Hu theorem for packing odd trails

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