Related papers: The $d$-dimensional realisation number of a rigid …
It is shown that $N$ points on a real algebraic curve of degree $n$ in $\mathbb{R}^d$ always determine $\gtrsim_{n,d}N^{1+\frac{1}{4}}$ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the…
Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…
Real-world graphs often manifest as a massive temporal stream of edges. The need for real-time analysis of such large graph streams has led to progress on low memory, one-pass streaming graph algorithms. These algorithms were designed for…
We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these…
We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on $2n$ vertices are bounded by…
We continue the study of the area requirement of convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by applications, such as graph editors, we additionally…
We show that determining if an $n$-vertex graph has twin-width at most 4 is NP-complete, and requires time $2^{\Omega(n/\log n)}$ unless the Exponential-Time Hypothesis fails. Along the way, we give an elementary proof that $n$-vertex…
We show that planar embeddable 3-connected CAD graphs are generically non-soluble. A CAD graph represents a configuration of points on the Euclidean plane with just enough distance dimensions between them to ensure rigidity. Formally, a CAD…
We are given an integer $d$, a graph $G=(V,E)$, and a uniformly random embedding $f : V \rightarrow \{0,1\}^d$ of the vertices. We are interested in the probability that $G$ can be "realized" by a scaled Euclidean norm on $\mathbb{R}^d$, in…
A measure for the visual complexity of a straight-line crossing-free drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph $G$, the minimum such number (over all drawings in dimension $d \in…
The d-Cut problem is to decide if a graph has an edge cut such that each vertex has at most d neighbours at the opposite side of the cut. If $d=1$, we obtain the intensively studied Matching Cut problem. The d-Cut problem has been studied…
The metric dimension (MD) of a graph is a combinatorial notion capturing the minimum number of landmark nodes needed to distinguish every pair of nodes in the graph based on graph distance. We study how much the MD can increase if we add a…
Many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbours to each point of a dataset to perform their tasks. These proximity relations define a so-called…
A construction of convex flag triangulations of five and higher dimensional spheres, whose h-polynomials fail to have only real roots, is given. We show that there is no such example in dimensions lower than five. A condition weaker than…
A good edge-labelling of a simple, finite graph is a labelling of its edges with real numbers such that, for every ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. In this paper we prove that any graph on…
The development of laser scanning techniques has popularized the representation of 3D shapes by triangular meshes with a large number of vertices. Compression techniques dedicated to such meshes have emerged, which exploit the idea that the…
Let G = (V, E) be a directed graph on n vertices where each vertex has out-degree k. We say that G is kNN-realizable in d-dimensional Euclidean space if there exists a point set P = {p1, p2, ..., pn} in R^d along with a one-to-one mapping…
The reachability problem asks to decide if there exists a path from one vertex to another in a digraph. In a grid digraph, the vertices are the points of a two-dimensional square grid, and an edge can occur between a vertex and its…
Motivated by a result of [1] which states that if F is a subgraph of a convex complete graph K_n and F contains no boundary edge of K_n and |E(F)| \leq n-3, then K_n - F admits a triangulation, we determine necessary and sufficient…
We study the maximum dimension $d=d(n,p)$ for which an Erd\H{o}s-R\'enyi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point…