Related papers: Unit Ball Graphs on Geodesic Spaces
A geodesic flower is a finite collection of geodesic loops based at the same point $p$ that satisfy the following balancing condition: The sum of all unit tangent vectors to all geodesic arcs meeting at $p$ is equal to the zero vector. In…
Graph embeddings, wherein the nodes of the graph are represented by points in a continuous space, are used in a broad range of Graph ML applications. The quality of such embeddings crucially depends on whether the geometry of the space…
A gain graph is a triple (G,h,H), where G is a connected graph with an arbitrary, but fixed, orientation of edges, H is a group, and h is a homomorphism from the free group on the edges of G to H. A gain graph is called balanced if the…
The primary objective of this paper is to investigate the notions of geometric and sequential convexity within a graph-theoretic framework, with the aim of examining various structural properties and exploring the connection between these…
Let $G$ be a connected graph and let $d_G$ be the geodesic distance on $V(G)$. The metric spaces $(V(G), d_{G})$ are characterized up to isometry for all finite connected $G$ by David C. Kay and Gary Chartrand in 1964. The main result of…
We investigate string graphs through the lens of graph product structure theory, which describes complicated graphs as subgraphs of strong products of simpler building blocks. A graph $G$ is called a string graph if its vertices can be…
For arbitrary semimetric space $(X, d)$ and disjoint proximinal subsets $A$, $B$ of $X$ we define the proximinal graph as a bipartite graph with parts $A$ and $B$ whose edges $\{a, b\}$ satisfy the equality $d(a, b) = \operatorname{dist}(A,…
We prove that a graph on up to 9 vertices is a unit-distance graph if and only if it does not contain one of 74 so-called minimal forbidden graphs. This extends the work of Chilakamarri and Mahoney (1995), who provide a similar…
A geodesic is a shortest path which connects a pair of vertices of a graph G. In this paper we define the geodesic subpath number gpn(G) of a graph G as the number of geodesics in G. The number of subtrees and subpaths are already studied…
A \emph{complete geometric graph} consists of a set $P$ of $n$ points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a…
In this paper we explore the idea that Teichm\"uller space is hyperbolic "on average." Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichm\"uller space. We…
In 1962, Oystein Ore asked in which graphs there is exactly one geodesic between any two vertices. He called such graphs geodetic. In this paper, we systematically study properties of geodetic graphs, and also consider antipodal graphs, in…
We study the geodesic complexity of the ordered and unordered configuration spaces of graphs in both the $\ell_1$ and $\ell_2$ metrics. We determine the geodesic complexity of the ordered two-point $\varepsilon$-configuration space of any…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…
Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such…
The strong geodetic number, $\text{sg}(G),$ of a graph $G$ is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the…
Given a family of graphs $\mathcal{H}$, a graph $G$ is $\mathcal{H}$-free if any subset of $V(G)$ does not induce a subgraph of $G$ that is isomorphic to any graph in $\mathcal{H}$. We present sufficient and necessary conditions for a graph…
A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In…
We characterise the slices of the category of graphs that are algebraically universal in terms of the structure of the slicing graph. In particular, we show that algebraic universality is obtained if, and only if, the slicing graph contains…
We show that the groupoids of two directed graphs are isomorphic if and only if the two graphs are orbit equivalent by an orbit equivalence that preserves isolated eventually periodic points. We also give a complete description of the…