Related papers: Decomposition of Binary Signed-Graphic Matroids

Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph…

We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs.

In this paper, a new general decomposition theory inspired from modular graph decomposition is presented. Our main result shows that, within this general theory, most of the nice algorithmic tools developed for modular decomposition are…

Let $\mathcal M$ be a class of matroids closed under minors and isomorphism. Let $N$ be a matroid in $\mathcal M$ with an exact $k$-separation $(A, B)$. We say $N$ is a $k$-decomposer for $\mathcal M$ having $(A, B)$ as an inducer, if every…

The effect of replacing a basis element on the way the basis spans other elements is studied. This leads to a new characterization of binary matroids.

We study the decomposition of 4d $\mathcal{N}=1$ gauge theories with Lie algebras of type $\mathfrak{su}(N)$, $\mathfrak{so}(2N)$, and $\mathfrak{e}_{6}$, realized via M-theory geometric engineering. These theories, together with their…

We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids.

The tree of decomposition of a $k$-connected graph by a set $\mathfrak S$ of pairwise independent $k$-vertex cutsets is defined as follows. The vertices of this tree are cutsets of $\mathfrak S$ and parts of decomposition of the graph by…

In Graph Theory a number of results were devoted to studying the computational complexity of the number modulo 2 of a graph's edge set decompositions of various kinds, first of all including its Hamiltonian decompositions, as well as the…

The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas proved that the base polytope of no binary matroid includes the base polytope of a connected matroid. A matroid base polytope is said to be…

In this paper we use 3-manifold techniques to illuminate the structure of the string link monoid. In particular, we give a prime decomposition theorem for string links on two components as well as give necessary conditions for string links…

We prove that there is no polynomial $p(\cdot)$ with the property that a matroid $M$ can be determined to be either a lifted-graphic or frame matroid using at most $p(|M|)$ rank evaluations. This resolves two conjectures of Geelen, Gerards…

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A…

Binary functions are a generalisation of the cocircuit spaces of binary matroids to arbitrary functions. Every rank function is assigned a binary function, and the deletion and contraction operations of binary functions generalise matroid…

Contraction$^*$-depth is considered to be one of the analogues of graph tree-depth in the matroid setting. In this paper, we investigate structural properties of contraction$^*$-depth of matroids representable over finite fields and…

In this paper Moussouris' algorithm for the decomposition of spin networks is reviewed and the implicit assumptions made in the Decomposition Theorem relating a spin network with its state sum are examined. It is found that the theorem in…

The handle slide operation, originally defined for ribbon graphs, was extended to delta-matroids by I. Moffatt and E. Mphako-Bandab, who show that, using a delta-matroid analogue of handle slides, every binary delta-matroid in which the…

Given graphs $G$ and $H$, we consider the problem of decomposing a properly edge-colored graph $G$ into few parts consisting of rainbow copies of $H$ and single edges. We establish a close relation to the previously studied problem of…

We prove that the non-regular binary matroids with no $P_9^*$-minor have linear growth rate and the maximum size binary matroids with no $P_9^*$-minor are graphic. The main technique in the proof is the Strong Splitter Theorem using which…

Let M be a compressionbody containing a graph T (with at least one edge) such that \boundary_+ M is parallel to the union of T and \boundary_- M. We extend methods of Hayashi and Shimokawa to classify bridge surfaces for T. The results of…