Related papers: On Visibility Graphs of Point Sets in the Plane

An \emph{obstacle representation} of a graph consists of a set of polygonal obstacles and a distinct point for each vertex such that two points see each other if and only if the corresponding vertices are adjacent. Obstacle representations…

We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results…

A geometric graph is a drawing of a graph in the plane where the vertices are drawn as points in general position and the edges as straight-line segments connecting their endpoints. It is plane if it contains no crossing edges. We study…

Given a graph $G$, a mutual-visibility coloring of $G$ is introduced as follows. We color two vertices $x,y\in V(G)$ with a same color, if there is a shortest $x,y$-path whose internal vertices have different colors than $x,y$. The smallest…

The point-thickness $\theta'(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ of $G$ is partitioned such that each subset induces a planar subgraph. In this paper, we determine the point-thickness of…

A non-zero component graph $G(\mathbb{V})$ associated to a finite vector space $\mathbb{V}$ is a graph whose vertices are non-zero vectors of $\mathbb{V}$ and two vertices are adjacent, if their corresponding vectors have at least one…

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part…

Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained…

Let $W(G)$ be the Wiener index of a graph $G$. We say that a vertex $v \in V(G)$ is a \v{S}olt\'es vertex in $G$ if $W(G - v) = W(G)$, i.e. the Wiener index does not change if the vertex $v$ is removed. In 1991, \v{S}olt\'es posed the…

A lattice point $\vec x=(x_1,\dots,x_n)\in\mathbb Z^{n}$ is said to be visible if the line segment between $\vec x$ and the origin contains no other lattice point. In this paper, we compute the asymptotic density of visible lattice points…

We introduce and study the pinnacle sets of a simple graph $G$ with $n$ vertices. Given a bijective vertex labeling $\lambda\,:\,V(G)\rightarrow [n]$, the label $\lambda(v)$ of vertex $v$ is a pinnacle of $(G, \lambda)$ if…

For a finite set $X$ of points in the plane, a set $S$ in the plane, and a positive integer $k$, we say that a $k$-element subset $Y$ of $X$ is captured by $S$ if there is a homothetic copy $S'$ of $S$ such that $X\cap S' = Y$, i.e., $S'$…

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…

We consider a set of cliques in any multipartite graph with two vertices in each part. Moreover, we construct a class of peculiar polytopes. Key words: multipartite graph, clique, polytope.

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: L, \Gamma, LE{} and \eeG. A $k$-bend path is a simple path in the plane, whose direction changes $k$ times from horizontal…

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…

The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions:…

A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small…

We consider a generalization of the Gabriel graph, the witness Gabriel graph. Given a set of vertices P and a set of witnesses W in the plane, there is an edge ab between two points of P in the witness Gabriel graph GG-(P,W) if and only if…

Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$…