Related papers: Dynamic Connectivity in Disk Graphs

Let $S \subseteq \mathbb{R}^2$ be a set of $n$ \emph{sites} in the plane, so that every site $s \in S$ has an \emph{associated radius} $r_s > 0$. Let $D(S)$ be the \emph{disk intersection graph} defined by $S$, i.e., the graph with vertex…

The intersection graph induced by a set $\Disks$ of $n$ disks can be dense. It is thus natural to try and sparsify it, while preserving connectivity. Unfortunately, sparse graphs can always be made disconnected by removing a small number of…

Let $\mathcal{D}$ be a set of $n$ disks in the plane. The disk graph $G_\mathcal{D}$ for $\mathcal{D}$ is the undirected graph with vertex set $\mathcal{D}$ in which two disks are joined by an edge if and only if they intersect. The…

A classical problem in computational geometry and graph algorithms is: given a dynamic set S of geometric shapes in the plane, efficiently maintain the connectivity of the intersection graph of S. Previous papers studied the setting where,…

Let $S \subset \mathbb{R}^2$ be a set of $n$ sites, where each $s \in S$ has an associated radius $r_s > 0$. The disk graph $D(S)$ is the undirected graph with vertex set $S$ and an undirected edge between two sites $s, t \in S$ if and only…

Let $P$ be a set of $n\geq 3$ points in general position in the plane. The edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in…

We present the first fully dynamic connectivity data structures for geometric intersection graphs achieving constant query time and sublinear amortized update time for most types of geometric objects in 2D. Our data structures can answer…

A disk graph is the intersection graph of disks in the plane, a unit disk graph is the intersection graph of same radius disks in the plane, and a segment graph is an intersection graph of line segments in the plane. It can be seen that…

In this paper, we present new incremental algorithms for maintaining data structures that represent all connectivity cuts of size one in directed graphs (digraphs), and the strongly connected components that result by the removal of each of…

This paper addresses the problem of finding an orientation and a minimum radius for directional antennas of a fixed angle placed at the points of a planar set S, that induce a strongly connected communication graph. We consider problem…

Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions.…

Given a convex polygon of $n$ sides, one can draw $n$ disks (called side disks) where each disk has a different side of the polygon as diameter and the midpoint of the side as its center. The intersection graph of such disks is the…

Given a finite set $ S $ of points, we consider the following reconfiguration graph. The vertices are the plane spanning paths of $ S $ and there is an edge between two vertices if the two corresponding paths differ by two edges (one…

Disk contact representations realize graphs by mapping vertices bijectively to interior-disjoint disks in the plane such that two disks touch each other if and only if the corresponding vertices are adjacent in the graph. Deciding whether a…

The vertex connectivity of a graph $G$ is the size of the smallest set of vertices $S$ such that $G \setminus S$ is disconnected. For the class of planar graphs, the problem of vertex connectivity is well-studied, both from structural and…

Let $G=(V, E)$ be a planar graph and let $\mathcal{C}$ be a partition of $V$. We refer to the graphs induced by the vertex sets in $\mathcal{C}$ as Clusters. Let $D_{\mathcal C}$ be an arrangement of disks with a bijection between the disks…

Let $S \subset \mathbb{R}^2$ be a set of $n$ sites. The unit disk graph $\text{UD}(S)$ on $S$ has vertex set $S$ and an edge between two distinct sites $s,t \in S$ if and only if $s$ and $t$ have Euclidean distance $|st| \leq 1$. A routing…

Temporal graphs represent interactions between entities over time. These interactions may be direct, a contact between two vertices at some time instant, or indirect, through sequences of contacts called journeys. Deciding whether an entity…

Let $V\subset\mathbb{R}^2$ be a set of $n$ sites in the plane. The unit disk graph $DG(V)$ of $V$ is the graph with vertex set $V$ in which two sites $v$ and $w$ are adjacent if and only if their Euclidean distance is at most $1$. We…

Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst-case guarantees. We propose a new data…