Related papers: Shortest Path Separators in Unit Disk Graphs

We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension $d$. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance…

This paper develops a structural theory of unique shortest paths in real-weighted graphs. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can be realized as unique shortest paths in a graph…

We study graph classes modeled by families of non-crossing (NC) connected sets. Two classic graph classes in this context are disk graphs and proper interval graphs. We focus on the cases when the sets are paths and the host is a tree…

Fat minors are a coarse analogue of graph minors where the subgraphs modeling vertices and edges of the embedded graph are required to be distant from each other, instead of just being disjoint. In this paper, we give a coarse analogue of…

Let $G=(V,E,w)$ be a weighted directed graph without negative cycles. For two vertices $s,t\in V$, we let $d_{\le h}(s,t)$ be the minimum, according to the weight function $w$, of a path from $s$ to $t$ that uses at most $h$ edges, or hops.…

We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too…

Let $G$ be a graph on $n$ vertices, $p$ the order of a longest path and $\kappa$ the connectivity of $G$. In 1989, Bauer, Broersma Li and Veldman proved that if $G$ is a 2-connected graph with $d(x)+d(y)+d(z)\ge n+\kappa$ for all triples…

In SoCG 2022, Conroy and T\'oth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an $O(n\log n)$-size 3-hop spanner for $n$ disks (or fat convex objects) in the plane, and an $O(n\log^2…

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…

In this paper, our goal is to characterize two graph classes based on the properties of minimal vertex (edge) separators. We first present a structural characterization of graphs in which every minimal vertex separator is a stable set. We…

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete…

We present a $O(n^{\frac{3}{2}})$-time algorithm for the \emph{shortest (diagonal) flip path problem} for \emph{lattice} triangulations with $n$ points, improving over previous $O(n^2)$-time algorithms. For a large, natural class of inputs,…

In this work we study shortest path problems in multimode graphs, a generalization of the min-distance measure introduced by Abboud, Vassilevska W. and Wang in [SODA'16]. A multimode shortest path is the shortest path using one of multiple…

We consider the problem of finding the minimum-weight subgraph that satisfies given connectivity requirements. Specifically, given a requirement $r \in \{0,1,2,3\}$ for every vertex, we seek the minimum-weight subgraph that contains, for…

We introduce the notion of a graph derangement, which naturally interpolates between perfect matchings and Hamiltonian cycles. We give a necessary and sufficient condition for the existence of graph derangements on a locally finite graph.…

A drawback of the classic approach for complexity analysis of distributed graph problems is that it mostly informs about the complexity of notorious classes of ``worst case'' graphs. Algorithms that are used to prove a tight (existential)…

Given a directed graph, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even…

Graphs are fundamental objects that find widespread applications across computer science and beyond. Graph Theory has yielded deep insights about structural properties of various families of graphs, which are leveraged in the design and…

The notion of shortcut partition, introduced recently by Chang, Conroy, Le, Milenkovi\'c, Solomon, and Than [CCLMST23], is a new type of graph partition into low-diameter clusters. Roughly speaking, the shortcut partition guarantees that…

Given a graph and a pair of terminals $s$, $t$, the next-to-shortest path problem asks for an $s\!\to \!t$ (simple) path that is shortest among all not shortest $s\!\to \!t$ paths (if one exists). This problem was introduced in 1996, and…