Integral Polyhedra Related to Even-Cycle and Even-Cut Matroids

Abstract

A family of sets ℋ is ideal if the polyhedron {x ≥ 0 : Σi∈Sxi ≥ 1, ∀S ∈ ℋ} is integral. Consider a graph G with vertices s, t. An odd st-walk is either an odd st-path or the union of an even st-path and an odd circuit that share, at most, one vertex. Let T be a subset of vertices of even cardinality. An st-T-cut is a cut of the form δ(U) where |U ∩ T| is odd and U contains exactly one of s or t. We give excluded minor characterizations for when the families of odd st-walks and st-T-cuts (represented as sets of edges) are ideal. As a corollary, we characterize which extensions and coextensions of graphic and cographic matroids are 1-flowing.

The Cycling Property for the Clutter of Odd st-Walks

“…A signed graph is said to be weakly bipartite if the clutter of its odd circuits is ideal. The clutter of odd circuits does not contain an L 7 minor [6]. Hence, we get the following two results as corollaries of Theorem 1.2: Corollary 2.2 (Guenin [5]).…”

“…As is shown in [6] the clutter of odd st-walks is binary, and it does not have a minor isomorphic to b(O 5 ) or P 10 . In this paper, we verify the Cycling Conjecture for this class of binary clutters: Theorem 1.2.…”

“…Corollary 2.1 (Guenin [6]). A clutter of odd st-walks is ideal if, and only if, it has no O 5 and no L 7 minor.…”

“…Otherwise, an even st-path P is carefully chosen and contracted to identify s and t and then the same lemma comes to the rescue, see Figure 3.7. In Part (6) we assume that two of L 1 , L 2 , L 3 are non-simple. In…”

Packing odd T‐joins with at most two terminals

“…(1) there exists a blocking vertex, (2) s, t is a blocking pair, (3) every minimal odd st-join is connected, (4) G is a plane graph with at most two odd faces, (5) G is a plane graph and u, v is a blocking pair where s, u, t, v appear on a facial cycle in this order, (6) G has an embedding on the projective plane where every face is even and s, t are connected by an odd edge.…”