Related papers: Non-crossing paths with geographic constraints
We consider straight line drawings of a planar graph $G$ with possible edge crossings. The \emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum…
Let $OWRN = \left\langle W_x,W_y \right\rangle$ be a One Way Road Network where $W_x$ and $W_y$ are the sets of directed horizontal and vertical roads respectively. $OWRN$ can be considered as a variation of directed grid graph. The…
For all $k \geq 1$, we show that deciding whether a graph is $k$-planar is NP-complete, extending the well-known fact that deciding 1-planarity is NP-complete. Furthermore, we show that the gap version of this decision problem is…
Given a set of obstacles and two points, is there a path between the two points that does not cross more than $k$ different obstacles? This is a fundamental problem that has undergone a tremendous amount of work. It is known to be NP-hard,…
Let $P$ be a set of $n \geq 5$ points in convex position in the plane. The path graph $G(P)$ of $P$ is an abstract graph whose vertices are non-crossing spanning paths of $P$, such that two paths are adjacent if one can be obtained from the…
Roughly speaking, gerrymandering is the systematic manipulation of the boundaries of electoral districts to make a specific (political) party win as many districts as possible. While typically studied from a geographical point of view,…
We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class…
It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the smallest…
Given an undirected graph $G=(V,E)$ and vertices $s,t,w_1,w_2\in V$, we study finding whether there exists a simple path $P$ from $s$ to $t$ such that $w_1,w_2 \in P$. As a sub-problem, we study the question: given an undirected graph and…
A monotone drawing of a planar graph $G$ is a planar straight-line drawing of $G$ where a monotone path exists between every pair of vertices of $G$ in some direction. Recently monotone drawings of planar graphs have been proposed as a new…
In this paper we study a path-following problem on $R^3$ with a non-holonomic constraint. The geometric structure associated to the velocity constraint is explored, and general principles for constructing guiding vector fields are obtained,…
The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It…
The directed landscape is a random directed metric on the plane that arises as the scaling limit of metric models in the KPZ universality class. For a pair of points p, q, the disjointness gap G(p; q) measures the shortfall when we optimize…
Grouping the nodes of a graph into clusters is a standard technique for studying networks. We study a problem where we are given a directed network and are asked to partition the graph into a sequence of coherent groups. We assume that…
A hypergraph consists of a set of vertices and a set of subsets of vertices, called hyperedges. In the metro map metaphor, each hyperedge is represented by a path (the metro line) and the union of all these paths is the support graph (metro…
A set of vertices X of a graph G is convex if it contains all vertices on shortest paths between vertices of X. We prove that for fixed p, all partitions of the vertex set of a bipartite graph into p convex sets can be found in polynomial…
A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We…
We consider questions related to the existence of spanning trees in graphs with the property that after the removal of any path in the tree the graph remains connected. We show that, for planar graphs, the existence of trees with this…
Perfect Matching-Cut is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for…
In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…