Related papers: Throttling for metric dimension and its variants
Threshold graphs are recursive deterministic network models that have been proposed for describing certain economic and social interactions. One drawback of this graph family is that it has limited generative attachment rules. To mitigate…
An obstacle representation of a graph $G$ consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of $G$ to points such that two vertices are adjacent in $G$ if and only if the line…
A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. Let $\{G_1, G_2, \ldots,…
The metric dimension, $\dim(G)$, of a graph $G$ is a graph parameter motivated by robot navigation that has been studied extensively. Let $G$ be a graph with vertex set $V(G)$, and let $d(x,y)$ denote the length of a shortest $x-y$ path in…
The Planar Graph Metric Compression Problem is to compactly encode the distances among $k$ nodes in a planar graph of size $n$. Two na\"ive solutions are to store the graph using $O(n)$ bits, or to explicitly store the distance matrix with…
Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$ we show…
By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…
The parametric shortest path problem is to find the shortest paths in graph where the edge costs are of the form w_ij+lambda where each w_ij is constant and lambda is a parameter that varies. The problem is to find shortest path trees for…
Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at…
Metric embeddings into structured spaces, particularly hierarchically well-separated trees (HSTs), are a fundamental tool in the design of online algorithms. In the classical online embedding setting, points arrive sequentially and must be…
The metric dimension of a graph measures how uniquely vertices may be identified using a set of landmark vertices. This concept is frequently used in the study of network architecture, location-based problems and communication. Given a…
We study the algorithmic aspect of edge bundling. A bundled crossing in a drawing of a graph is a group of crossings between two sets of parallel edges. The bundled crossing number is the minimum number of bundled crossings that group all…
In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like ``pegging moves'' allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging…
In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these…
Let $G=(V,E)$ be a connected graph, let $v\in V$ be a vertex and let $e=uw\in E$ be an edge. The distance between the vertex $v$ and the edge $e$ is given by $d_G(e,v)=\min\{d_G(u,v),d_G(w,v)\}$. A vertex $w\in V$ distinguishes two edges…
Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with…
We consider the problem of adding a fixed number of new edges to an undirected graph in order to minimize the diameter of the augmented graph, and under the constraint that the number of edges added for each vertex is bounded by an integer.…
Let G = (V, E) be a finite simple connected graph. We say a graph G realizes a code of the type 0^s_1 1^t_1 0^s_2 1^t_2 ... 0^s_k1^t_k if and only if G can obtained from the code by some rule. Some classes of graphs such as threshold and…
A metric graph is a 1-dimensional stratified metric space consisting of vertices and edges or loops glued together. Metric graphs can be naturally used to represent and model data that take the form of noisy filamentary structures, such as…
In this paper, we give approximation algorithms for the \textsc{Minimum Dominating Set (MDS)} problem on \emph{string} graphs and its subclasses. A \emph{path} is a simple curve made up of alternating horizontal and vertical line segments.…