Related papers: Realizing Planar Linkages in Polygonal Domains

A plane graph is said to be a rectangular graph if each of its edges can be oriented horizontal or vertical, its internal regions are four-sided and it has a rectangular enclosure. If dual of a planar graph is a rectangular graph, then the…

The algorithm of Gutwenger et al. to insert an edge $e$ in linear time into a planar graph $G$ with a minimal number of crossings on $e$, is a helpful tool for designing heuristics that minimize edge crossings in drawings of general graphs.…

Hierarchical embedding constraints define a set of allowed cyclic orders for the edges incident to the vertices of a graph. These constraints are expressed in terms of FPQ-trees. FPQ-trees are a variant of PQ-trees that includes F-nodes in…

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…

An ortho-polygon visibility representation of an $n$-vertex embedded graph $G$ (OPVR of $G$) is an embedding-preserving drawing of $G$ that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal…

A $d$-dimensional framework is a pair $(G,p)$, where $G=(V,E)$ is a graph and $p$ is a map from $V$ to $\mathbb{R}^d$. The length of an edge $xy\in E$ in $(G,p)$ is the distance between $p(x)$ and $p(y)$. A vertex pair $\{u,v\}$ of $G$ is…

A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked…

For a subset $X$ of the vertex set $\VV(\GG)$ of a graph $\GG$, we denote the set of edges of $\GG$ which have exactly one end in $X$ by $\partial(X)$ and refer to it as the cut of $X$ or edge cut $\partial(X)$. A graph $\GG=(\VV,\EE)$ is…

This paper studies the configuration space of all possible positions of a linkage in R^n. For example, it shows that for every compact algebraic set, there is a linkage whose configuration space is analytically isomorphic to a finite number…

In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph $G(V,E)$ and a function $\gamma:V \rightarrow \{1,2,\dots,k\}$ and asks whether a planar drawing of $G$ exists such that each…

We show that planar embeddable 3-connected CAD graphs are generically non-soluble. A CAD graph represents a configuration of points on the Euclidean plane with just enough distance dimensions between them to ensure rigidity. Formally, a CAD…

Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs…

We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we…

In this paper, we study fan-planar drawings that use $h$ layers and are proper, i.e., edges connect adjacent layers. We show that if the embedding of the graph is fixed, then testing the existence of such drawings is fixed-parameter…

A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain…

A $d$-dimensional (bar-and-joint) framework $(G,p)$ consists of a graph $G=(V,E)$ and a realisation $p:V\to \mathbb{R}^d$. It is rigid if every continuous motion of the vertices which preserves the lengths of the edges is induced by an…

The metric dimension of a graph $G$ is the size of a smallest subset $L \subseteq V(G)$ such that for any $x,y \in V(G)$ with $x\not= y$ there is a $z \in L$ such that the graph distance between $x$ and $z$ differs from the graph distance…

Let $G$ be a $3$-connected ordered graph with $n$ vertices and $m$ edges. Let $\mathbf{p}$ be a randomly chosen mapping of these $n$ vertices to the integer range $\{1, 2,3, \ldots, 2^b\}$ for $b\ge m^2$. Let $\ell$ be the vector of $m$…

Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we…

The \textsc{$l$-Exact Component Order Connectivity} problem asks whether, given an input graph $G$ and an integer $k$, there exists a vertex subset $S\subseteq V(G)$ of size at most $k$ such that every connected component in $G - S$ has…