Related papers: Vertex Collapsing and Cut Ideals
Kautz and de Bruijn graphs have a high degree of connectivity which makes them ideal candidates for massively parallel computer network topologies. In order to realize a practical computer architecture based on these graphs, it is useful to…
We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{|E(G)|},…
A partial edge drawing (PED) of a graph is a variation of a node-link diagram. PED draws a link, which is a partial visual representation of an edge, and reduces visual clutter of the node-link diagram. However, more time is required to…
Throughout this work, the vertex decomposability and shellability of graphs formed from other graphs by various operations are investigated. Also among the other things, by using some graph operations, new classes of Cohen-Macaulay graphs…
The total irregularity of a graph $G$ is defined as $\irr_t(G)=1/2 \sum_{u,v \in V(G)}$ $|d_G(u)-d_G(v)|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. In this paper we give (sharp) upper bounds on the total irregularity of…
We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs…
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal…
Let $D$ be a weighted oriented graph with the underlying graph $G$ and $I(D), I(G) $ be the edge ideals corresponding to $D$ and $G$ respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph…
We investigate the geometric properties of loops on two-dimensional lattice graphs, where edge weights are drawn from a distribution that allows for positive and negative weights. We are interested in the appearance of spanning loops of…
The v-number of a graded ideal is an invariant recently introduced in the context of coding theory, particularly in the study of Reed--Muller-type codes. In this work, we study the localized v-numbers of a binomial edge ideal $J_G$…
Editing a graph into a disjoint union of clusters is a standard optimization task in graph-based data clustering. Here, complementing classic work where the clusters shall be cliques, we focus on clusters that shall be 2-clubs, that is,…
There is a graph reduction system so that every optimal 1-planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if…
Graph neural networks (GNNs) have become increasingly popular for classification tasks on graph-structured data. Yet, the interplay between graph topology and feature evolution in GNNs is not well understood. In this paper, we focus on…
We first characterise graphs with binomial edge ideals of K\"onig type as those for which the path covering number is equal to a minor variant of the scattering number. These are well-studied graph-theoretic invariants, allowing us to apply…
This is a new and short proof of the main theorem of classical structure tree theory. Namely, we show the existence of certain automorphism-invariant tree-decompositions of graphs based on the principle of removing finitely many edges. This…
We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…
With the advent of the big data, graph are processed in an iterative manner, which incrementally described in the form of graph in big data applications. Most currently, graph processing methods treat the underlying map data as black boxes.…
A graph is a mathematical object consisting of a set of vertices and a set of edges connecting vertices. Graphs can be drawn on paper in various ways, but until recently all published methods of drawing graphs have had undesirable…
This short note solves the following problem: Given a map of normal toric varieties corresponding to a coherent subdivision of a cone, find an ideal such that the given map is the blowup of that ideal.
Motivated by Kuratowski's theorem, a Kuratowski subgraph of a graph is a subgraph that is a subdivided $K_5$ or a subdivided $K_{3,3}$. An edge is crossing-critical if the crossing number decreases after removing the edge. In this note, we…