Related papers: A Graph Theory of Rook Placements
We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex…
A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H…
We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently…
Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column. This is the square rook's graph, and can also be thought of as…
A vertex with neighbours of degrees $d_1 \geq ... \geq d_r$ has {\em vertex type} $(d_1, ..., d_r)$. A graph is {\em vertex-oblique} if each vertex has a distinct vertex-type. While no graph can have distinct degrees, Schreyer, Walther and…
Let the class A of graphs be bridge-addable; that is, whenever a graph G in A has vertices u and v in different components then the graph G+uv is in A. For a random graph sampled uniformly from the graphs in A on vertex set {1,..,n}, there…
Given a set of points in the plane, we want to establish a connection network between these points that consists of several disjoint layers. Motivated by sensor networks, we want that each layer is spanning and plane, and that no edge is…
We say that a graph $F$ can be embedded into a graph $G$ if $G$ contains an isomorphic copy of $F$ as a subgraph. Guo and Volkmann \cite{GV} conjectured that if $G$ is a connected graph with at least $n$ vertices and minimum degree at least…
We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra. We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we…
A tangled-diagram over $[n]=\{1,...,n\}$ is a graph of degree less than two whose vertices $1,...,n$ are arranged in a horizontal line and whose arcs are drawn in the upper halfplane with a particular notion of crossings and nestings.…
We consider global dynamics of reaction systems as introduced by Ehrenfeucht and Rozenberg. The dynamics is represented by a directed graph, the so-called transition graph, and two reaction systems are considered equivalent if their…
We proved in another paper that every connected graph can be realized as the cut locus of some point on some riemannian surface. Here we give upper bounds on the number of such realizations.
A graph G is called (2k, k)-connected if G is 2k-edge-connected and G-v is k-edge-connected for every vertex v. The study of (2k, k)-connected graphs is motivated by a conjecture of Frank which states that a graph has a 2-vertex-connected…
A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open…
The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating…
We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree…
For a finite point set $E\subset \mathbb{R}^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k + 1$ points in E such that the distance between a pair of points is specified if the…
A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In…
The article deals with two classes of growing random graphs following the preferential attachment rule with a linear weight function, L-graphs, and hybrid Pennock graphs. We determine the exact final vertex degree distribution and the exact…
Roles represent node-level connectivity patterns such as star-center, star-edge nodes, near-cliques or nodes that act as bridges to different regions of the graph. Intuitively, two nodes belong to the same role if they are structurally…