Related papers: Unsolved Problems in Visibility Graphs of Points, …
Let G(V,E) be a simple graph and let X subset of V. Two vertices u and v are said to be X-visible if there exists a shortest u,v-path P such that V(P) intersection X is a subset of {u, v}. A set X is called a mutual-visibility set of G if…
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
We present some complexity results concerning the problems of covering a graph with $p$ convex sets and of partitioning a graph into $p$ convex sets. The following convexities are considered: digital convexity, monophonic convexity,…
The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi Riemannian and magnetic structures on manifolds.
A graph is said to be a segment graph if its vertices can be mapped to line segments in the plane such that two vertices have an edge between them if and only if their corresponding line segments intersect. Kratochv\'{i}l and Kub\v{e}na…
A segment representation of a graph is an assignment of line segments in 2D to the vertices in such a way that two segments intersect if and only if the corresponding vertices are adjacent. Not all graphs have such segment representations,…
A subgraph is constructed by using a subset of vertices and edges of a given graph. There exist many graph properties that are hereditary for subgraphs. Hence, researchers from different communities have paid a great deal of attention in…
We study the recently introduced problem of finding dense common subgraphs: Given a sequence of graphs that share the same vertex set, the goal is to find a subset of vertices $S$ that maximizes some aggregate measure of the density of the…
This is an exposition of results on the existence problem of $\pi_1$-injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds, obtained by different authors. The results are…
We define the problem segment cover as follows. We are given a set of pairs of sub-intervals of the unit interval. The problem asks if there is a choice of a single interval from each pair such that the union of the chosen intervals covers…
A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its…
Uncover the vertices of a given graph, deterministic or random, in random order; we consider both a discrete-time and a continuous-time version. We study the evolution of the number of visible edges, and show convergence after normalization…
Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering…
Existing approaches to solving combinatorial optimization problems on graphs suffer from the need to engineer each problem algorithmically, with practical problems recurring in many instances. The practical side of theoretical computer…
We consider the problem of mirror invisibility for plane sets. Given a circle and a finite number of unit vectors (defining the directions of invisibility) such that the angles between them are commensurable with $\pi$, for any $\varepsilon…
In this work, I use a survey of senior visualization researchers and thinkers to ideate about the notion of progress in visualization research: how are we growing as a field, what are we building towards, and are our existing methods…
Mutual-visibility sets were motivated by visibility in distributed systems and social networks, and intertwine with several classical mathematical areas. Monotone properties of the variety of mutual-visibility sets, and restrictions of such…
The notions of spectral measures and spectral classes, which are well known for graphs, are generalized and investigated for oriented hypergraphs.
Graphs represent interconnected structures prevalent in a myriad of real-world scenarios. Effective graph analytics, such as graph learning methods, enables users to gain profound insights from graph data, underpinning various tasks…
A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We…