Related papers: On $Z$-monodromies in embedded graphs

In this work, we prove a general version of the reduction lemmas for eigenfunctions of graphs admitting involutive automorphisms of a special type.

This paper introduces a novel topology, referred to as the star topology, on finite graphs. By treating vertices and edges as points in a unified space, we explore continuous maps between Bare representations of a graph and their…

Given a graph G, we construct a convex polytope whose face poset is based on marked subgraphs of G. Dubbed the graph multiplihedron, we provide a realization using integer coordinates. Not only does this yield a natural generalization of…

In this paper we consider del Pezzo surfaces with only log terminal singularities admitting an action of a finite simple group.

In this short note, we prove the uniqueness of exterior differentiation on locally finite graphs.

Generalizing results of Frucht and de Groot/Sabidussi, we demonstrate that every group-embeddable monoid is isomorphic to the bimorphism monoid of some graph.

We give an explicit calculation of the Wu invariants for immersions of a finite graph into the plane and classify all generic immersions of a graph into the plane up to regular homotopy by the Wu invariant. This result is a generalization…

We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the…

In an $\mathsf{L}$-embedding of a graph, each vertex is represented by an $\mathsf{L}$-segment, and two segments intersect each other if and only if the corresponding vertices are adjacent in the graph. If the corner of each…

A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected…

The mutually enriching relationship between graphs and matroids has motivated discoveries in both fields. In this paper, we exploit the similar relationship between embedded graphs and delta-matroids. There are well-known connections…

Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$…

Cographs--defined most simply as complete graphs with colored lines--both dualize and generalize ordinary graphs, and promise a comparably wide range of applications. This article introduces them by examples, catalogues, and elementary…

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and gamma-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra,…

$2$-stratifolds are a generalization of $2$-manifolds that occur as objects in applications such as in TDA. These spaces can be described by an associated bicoloured labelled graph. In previous papers we obtained a classification of…

Let $G$ be a finite simple graph with edge ideal $I(G)$. Let $J(G)$ denote the Alexander dual of $I(G)$. We show that a description of all induced cycles of odd length in $G$ is encoded in the associated primes of $J(G)^2$. This result…

A connected 3-valent plane graph, whose faces are $q$- or 6-gons only, is called a {\em graph $q_n$}. We classify all graphs $4_n$, which are isometric subgraphs of a $m$-hypercube $H_m$.

In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and…

We characterize which automorphisms of an arbitrary complete bipartite graph $K_{n,m}$ can be induced by a homeomorphism of some embedding of the graph in $S^3$.

We consider seven fundamental properties of cellular embeddings of graphs in compact surfaces, and show that each property can be associated with a point of the Fano plane $F$, in such a way that allowable combinations of properties…