Related papers: Cell structure of bipartite mediangle graphs
Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via…
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges…
In this note, we introduce a family of bipartite graphs called path restricted ordered bipartite graphs and present it as an abstract generalization of some well known geometric graphs like unit distance graphs on convex point sets. In the…
Mirror graphs were introduced by Bre\v{s}ar et al. in 2004 as an intriguing class of graphs: vertex-transitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article…
We study the class of 1-perfectly orientable graphs, that is, graphs having an orientation in which every out-neighborhood induces a tournament. 1-perfectly orientable graphs form a common generalization of chordal graphs and circular arc…
A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed…
Delta-matroid theory is often thought of as a generalization of topological graph theory. It is well-known that an orientable embedded graph is bipartite if and only if its Petrie dual is orientable. In this paper, we first introduce the…
Each finite and connected bipartite graph induces a finite collection of non-isomorphic dessins d'enfants, that is, $2$-cell embeddings of it into some closed orientable surface. We describe an algorithm to compute all these dessins…
We characterize which systems of sign vectors are the cocircuits of an oriented matroid in terms of the cocircuit graph.
We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third…
The median of a graph $G$ with weighted vertices is the set of all vertices $x$ minimizing the sum of weighted distances from $x$ to the vertices of $G$. For any integer $p\ge 2$, we characterize the graphs in which, with respect to any…
Oriented graph complexes, in which graphs are not allowed to have oriented cycles, govern for example the quantization of Lie bialgebras and infinite dimensional deformation quantization. It is shown that the oriented graph complex GC^or_n…
In a closed 2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some…
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…
The active bijection forms a package of results studied by the authors in a series of papers in oriented matroids. The present paper is intended to state the main results in the particular case, and more widespread language, of graphs. We…
We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible…
We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment.…
We introduce \textit{dual graph diagrams} representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures we call \textit{biquasiles} whose axioms are motivated by dual graph…
In this paper we give a method to construct Heegaard splittings of oriented graph manifolds with orientable bases. A graph manifold is a closed $3$-manifold admitting only Seifert-fibered pieces in its Jaco-Shalen decomposition; for…
We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with…