Packing odd TT-joins with at most two terminals

Take a graph $G$, an edge subset $\Sigma\subseteq E(G)$, and a set of terminals $T\subseteq V(G)$ where $|T|$ is even. The triple $(G,\Sigma,T)$ is called a signed graft. A $T$-join is odd if it contains an odd number of edges from $\Sigma$. Let $\nu$ be the maximum number of edge-disjoint odd $T$-joins. A signature is a set of the form $\Sigma\triangle \delta(U)$ where $U\subseteq V(G)$ and $|U\cap T)$ is even. Let $\tau$ be the minimum cardinality a $T$-cut or a signature can achieve. Then $\nu\leq \tau$ and we say that $(G,\Sigma,T)$ packs if equality holds here. We prove that $(G,\Sigma,T)$ packs if the signed graft is Eulerian and it excludes two special non-packing minors. Our result confirms the Cycling Conjecture for the class of clutters of odd $T$-joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two-commodity paths, packing $T$-joins with at most four terminals, and a new result on covering edges with cuts.Comment: extended abstract appeared in IPCO 2014 (under the different title "the cycling property for the clutter of odd st-walks"

Isomorphism for even cycle matroids - I

A seminal result by Whitney describes when two graphs have the same cycles. We consider the analogous problem for even cycle matroids. A representation of an even cycle matroid is a pair formed by a graph together with a special set of edges of the graph. Such a pair is called a signed graph. We consider the problem of determining the relation between two signed graphs representing the same even cycle matroid. We refer to this problem as the Isomorphism Problem for even cycle matroids. We present two classes of signed graphs and we solve the Isomorphism Problem for these two classes. We conjecture that, up to simple operations, any two signed graphs representing the same even cycle matroid are either in one of these classes, or related by a modification of an operation for graphic matroids, or belonging to a small set of examples

Clean clutters and dyadic fractional packings

A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary cas

Dyadic linear programming and extensions

A rational number is dyadic if it has a finite binary representation $p/2^k$, where $p$ is an integer and $k$ is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case

A Characterization Of Weakly Bipartite Graphs (Extended Abstract)

A labeled graph is said to be weakly bipartite if the clutter of its odd cycles is ideal. Seymour conjectured that a labeled graph is weakly bipartite if and only if it does not contain a minor called an odd K5 . A proof of this conjecture is given in this paper

The two-point Fano and ideal binary clutters

Let F be a binary clutter. We prove that if F is non-ideal, then either F or its blocker b(F) has one of L 7 , O 5 , LC 7 as a minor. L 7 is the non-ideal clutter of the lines of the Fano plane, O 5 is the non-ideal clutter of odd circuits of the complete graph K 5 , and the two-point FanoLC 7 is the ideal clutter whose sets are the lines, and their complements, of the Fano plane that contain exactly one of two fixed points. In fact, we prove the following stronger statement: if F is a minimally non-ideal binary clutter different from L 7 , O 5 , b(O 5 ) , then through every element, either F or b(F) has a two-point Fano minor

Ideal Binary Clutters, Connectivity, and a Conjecture Of Seymour

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 st-paths, st-cuts, T-joins or T-cuts 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 T-joins and T-cuts