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Decomposition, approximation, and coloring of oddminorfree graphs
"... We prove two structural decomposition theorems about graphs excluding a fixed odd minor H, and show how these theorems can be used to obtain approximation algorithms for several algorithmic problems in such graphs. Our decomposition results provide new structural insights into oddHminorfree graph ..."
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Cited by 5 (2 self)
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We prove two structural decomposition theorems about graphs excluding a fixed odd minor H, and show how these theorems can be used to obtain approximation algorithms for several algorithmic problems in such graphs. Our decomposition results provide new structural insights into oddHminorfree graphs, on the one hand generalizing the central structural result from Graph Minor Theory, and on the other hand providing an algorithmic decomposition into two boundedtreewidth graphs, generalizing a similar result for minors. As one example of how these structural results conquer difficult problems, we obtain a polynomialtime 2approximation for vertex coloring in oddHminorfree graphs, improving on the previous O(V (H))approximation for such graphs and generalizing the previous 2approximation for Hminorfree graphs. The class of oddHminorfree graphs is a vast generalization of the wellstudied Hminorfree graph families and includes, for example, all bipartite graphs plus a bounded number of apices. OddHminorfree graphs are particularly interesting from a structural graph theory perspective because they break away from the sparsity of Hminorfree graphs, permitting a quadratic number of edges.
Graph minor algorithm with the parity condition
"... We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems: 1) the parity Hminor (Odd Kkminor) containment problem, and 2) the disjoint paths problem with k terminals and the parity condition for ea ..."
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We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems: 1) the parity Hminor (Odd Kkminor) containment problem, and 2) the disjoint paths problem with k terminals and the parity condition for each path, as well as several other related problems. We present an O(mα(m,n)n) time algorithm for these problems 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 Tarjan [69]). Note that the first problem includes the problem of testing whether or not a given graph contains k disjoint odd cycles
Strong tperfection of badK_4free graphs
 SIAM journal on Discrete Mathematics
, 2002
"... We show that each graph not containing a bad subdivision of K 4 as a subgraph, is strongly tperfect. Here a graph G = (V; E) is strongly tperfect if for each weight function w : V ! Z+ , the maximum weight of a stable set is equal to the minimum (total) cost of a family of vertices, edges, and cir ..."
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Cited by 3 (0 self)
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We show that each graph not containing a bad subdivision of K 4 as a subgraph, is strongly tperfect. Here a graph G = (V; E) is strongly tperfect if for each weight function w : V ! Z+ , the maximum weight of a stable set is equal to the minimum (total) cost of a family of vertices, edges, and circuits covering any vertex v at least w(v) times. By definition, the cost of a vertex or edge is 1, and the cost of a circuit C is b 1 2 jV Cjc. A subdivision of K 4 is called bad if each triangle has become an odd circuit and if it is not obtained by making the edges in a 4circuit of K 4 evenly subdivided, while the other two edges are not subdivided.
A Short Proof Of Seymour's Characterization Of The Matroids With The MaxFlow MinCut Property
"... Seymour proved that the set of odd circuits of a signed binary matroid (M;) has the MaxFlow MinCut property if and only if it does not contain a minor isomorphic to (M(K4);E(K4)). ..."
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Seymour proved that the set of odd circuits of a signed binary matroid (M;) has the MaxFlow MinCut property if and only if it does not contain a minor isomorphic to (M(K4);E(K4)).
PACKING ODD CIRCUITS
, 2007
"... We determine the structure of a class of graphs that do not contain the complete graph on five vertices as a “signed minor.” The result says that each graph in this class can be decomposed into elementary building blocks in which maximum packings by odd circuits can be found by flow or matching tech ..."
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We determine the structure of a class of graphs that do not contain the complete graph on five vertices as a “signed minor.” The result says that each graph in this class can be decomposed into elementary building blocks in which maximum packings by odd circuits can be found by flow or matching techniques. This allows us to actually find a largest collection of pairwise edge disjoint odd circuits in polynomial time (for general graphs this is NPhard). Furthermore it provides an algorithm to test membership of our class of graphs.
Packing Circuits in Matroids
, 2007
"... The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: For any nonnegative integral weight function w defined on E(M), Maximum {k: M has k circuits (repetition allowed) such that each element e of M is used at most 2w(e) times by these circuits} = Mi ..."
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The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: For any nonnegative integral weight function w defined on E(M), Maximum {k: M has k circuits (repetition allowed) such that each element e of M is used at most 2w(e) times by these circuits} = Minimum { � x∈X w(x) : X is a collection of elements (repetition allowed) of M such that every circuit in M meets X at least twice}. Our characterization contains a complete solution to a research problem on 2edgeconnected subgraph polyhedra posed by Cornuéjols, Fonlupt, and Naddef in 1985, which was independently solved by Vandenbussche and Nemhauser in [11].
Packing odd Tjoins with at most two terminals
, 2014
"... We prove the Cycling conjecture for the class of clutters of odd Tjoins with at most two terminals. Corollaries of this result include, the characterization of weakly bipartite graphs [5], packing twocommodity paths [7, 10], packing Tjoins for T  ≤ 4, and a new result on covering edges with cu ..."
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We prove the Cycling conjecture for the class of clutters of odd Tjoins with at most two terminals. Corollaries of this result include, the characterization of weakly bipartite graphs [5], packing twocommodity paths [7, 10], packing Tjoins for T  ≤ 4, and a new result on covering edges with cuts.
Displaying blocking pairs in signed graphs
, 2011
"... A signed graph is a pair (G,⌃) where G is a graph and ⌃ is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of ⌃. A blocking pair of (G,⌃) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In thi ..."
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A signed graph is a pair (G,⌃) where G is a graph and ⌃ is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of ⌃. A blocking pair of (G,⌃) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no oddK5 minor.