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Packing Odd Circuits in Eulerian Graphs
 JOURNAL OF COMBINATORIAL THEORY, SERIES B
, 2001
"... Let C be the clutter of odd circuits of a signed graph (G; ). For nonnegative integral edge{weights w, we are interested in the linear program min(w t x : x(C) 1; for C 2 C; and x 0), which we denote by (P ). Solving the related integer program is clearly equivalent to the maximum cut problem, ..."
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Let C be the clutter of odd circuits of a signed graph (G; ). For nonnegative integral edge{weights w, we are interested in the linear program min(w t x : x(C) 1; for C 2 C; and x 0), which we denote by (P ). Solving the related integer program is clearly equivalent to the maximum cut problem, which is NP{hard. Guenin proved that (P ) has an optimal solution that is integral so long as (G; ) does not contain a minor isomorphic to odd{K 5 . We generalize this by showing that, if (G; ) does not contain a minor isomorphic to odd{K 5 then (P ) has an integral optimal solution and its dual has a half{integral optimal solution.
Ideal Binary Clutters, Connectivity, and a Conjecture Of Seymour
, 2001
"... A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that intersect with odd cardinality a fixed given subset of elements. Let A denote the 0; 1 matrix whose rows are the characteristic vectors of the odd circuits. A binary clutter is ideal if the pol ..."
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Cited by 3 (1 self)
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A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that intersect with odd cardinality a fixed given subset of elements. Let A denote the 0; 1 matrix whose rows are the characteristic vectors of the odd circuits. A binary clutter is ideal if the polyhedron fx 0 : Ax 1g is integral. Examples of ideal binary clutters are stpaths, stcuts, Tjoins or Tcuts in graphs, and odd circuits in weakly bipartite graphs. In 1977, Seymour conjectured that a binary clutter is ideal if and only if it does not contain L F 7 , OK 5 , or b(O K 5 ) as a minor. In this paper, we show that a binary clutter is ideal if it does not contain five specified minors, namely the three above minors plus two others. This generalizes Guenin's characterization of weakly bipartite graphs, as well as the theorem of Edmonds and Johnson on Tjoins and Tcuts.
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 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.