Matroids arising from electrical networks

This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main results. First, we exhibit a matroid quotient formed by the dual of a network embedded in a surface with boundary and the dual of the associated Dirichlet matroid. This generalizes an analogous result for graphic matroids of cellularly embedded graphs. Second, we characterize the Bergman fans of Dirichlet matroids as explicit subfans of graphic Bergman fans. In doing so, we generalize the connection between Bergman fans of complete graphs and phylogenetic trees. Third, we use the half-plane property of Dirichlet matroids to prove an interlacing result on the real zeros and poles of the trace of the response matrix. And fourth, we bound the coefficients of the precoloring polynomial of a network by the coefficients of the chromatic polynomial of the underlying graph.Comment: 27 pages, 14 figure

Graphical representations of graphic frame matroids

A frame matroid M is graphic if there is a graph G with cycle matroid isomorphic to M. In general, if there is one such graph, there will be many. Zaslavsky has shown that frame matroids are precisely those having a representation as a biased graph; this class includes graphic matroids, bicircular matroids, and Dowling geometries. Whitney characterized which graphs have isomorphic cycle matroids, and Matthews characterised which graphs have isomorphic graphic bicircular matroids. In this paper, we give a characterization of which biased graphs give rise to isomorphic graphic frame matroids

Bicircular Matroids with Circuits of at Most Two Sizes

Young in his paper titled, Matroid Designs in 1973, reports that Murty in his paper titled, Equicardinal Matroids and Finite Geometries in 1968, was the first to study matroids with all hyperplanes having the same size. Murty called such a matroid an ``Equicardinal Matroid\u27\u27. Young renamed such a matroid a ``Matroid Design\u27\u27. Further work on determining properties of these matroids was done by Edmonds, Murty, and Young in their papers published in 1972, 1973, and 1970 respectively. These authors were able to connect the problem of determining the matroid designs with specified parameters with results on balanced incomplete block designs. The dual of a matroid design is one in which all circuits have the same size. In 1971, Murty restricted his attention to binary matroids and was able to characterize all connected binary matroids having circuits of a single size. Lemos, Reid, and Wu in 2010, provided partial information on the class of connected binary matroids having circuits of two different sizes. They also shothat there are many such matroids. In general, there are not many results that specify the matroids with circuits of just a few different sizes. Cordovil, Junior, and Lemos provided such results on matroids with small circumference. Here we determine the connected bicircular matroids with all circuits having the same size. We also provide structural information on the connected bicircular matroids with circuits of two different sizes. The bicircular matroids considered are in general non-binary. Hence these results are a start on extending Murty\u27s characterization of binary matroid designs to non-binary matroids

Polynomials with the half-plane property and matroid theory

A polynomial f is said to have the half-plane property if there is an open half-plane H, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions regarding multivariate polynomials with the half-plane property and matroid theory. * We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous. * We characterize multivariate multi-affine polynomial with real coefficients that have the half-plane property (with respect to the upper half-plane) in terms of inequalities. This is used to answer two open questions posed by Choe and Wagner regarding strongly Rayleigh matroids. * We prove that the Fano matroid is not the support of a polynomial with the half-plane property. This is the first instance of a matroid which does not appear as the support of a polynomial with the half-plane property and answers a question posed by Choe et al. We also discuss further directions and open problems.Comment: 17 pages. To appear in Adv. Mat

Finding Even Subgraphs Even Faster

Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on $n$ vertices and a positive integer parameter $k$, find if there exist $k$ edges (arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all the vertices have even degrees---where the resulting graph is \emph{Eulerian}---the problem is called Undirected Eulerian Edge Deletion. The corresponding problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its out-degree is called Directed Eulerian Edge Deletion. Cygan et al. [\emph{Algorithmica, 2014}] showed that these problems are fixed parameter tractable (FPT), and gave algorithms with the running time $2^{O(k \log k)}n^{O(1)}$. They also asked, as an open problem, whether there exist FPT algorithms which solve these problems in time $2^{O(k)}n^{O(1)}$. In this paper we answer their question in the affirmative: using the technique of computing \emph{representative families of co-graphic matroids} we design algorithms which solve these problems in time $2^{O(k)}n^{O(1)}$. The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid. We believe that this view-point/approach will be useful in other problems where one of the constraints that need to be satisfied is that of connectivity