We consider the problem of extending the drawing of a subgraph of a given plane graph to a drawing of the entire graph using straight-line and polyline edges. We define the notion of star complexity of a polygon and show that a drawing $Γ_H$ of an induced connected subgraph $H$ can be extended with at most $\min\{ h/2, β+ \log_2(h) + 1\}$ bends per edge, where $β$ is the largest star complexity of a face of $Γ_H$ and $h$ is the size of the largest face of $H$. This result significantly improves the previously known upper bound of $72|V(H)|$ [5] for the case where $H$ is connected. We also show that our bound is worst case optimal up to a small additive constant. Additionally, we provide an indication of complexity of the problem of testing whether a star-shaped inner face can be extended to a straight-line drawing of the graph; this is in contrast to the fact that the same problem is solvable in linear time for the case of star-shaped outer face [9] and convex inner face [13].
Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a common endpoint or at a proper crossing. In this paper we study properties of maximal plane subgraphs of simple drawings $D_n$ of the complete graph $K_n$ on $n$ vertices. Our main structural result is that maximal plane subgraphs are 2-connected and what we call essentially 3-edge-connected. Besides, any maximal plane subgraph contains at least $\lceil 3n/2 …
We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.
Given a graph $G$ and a subset $F \subseteq E(G)$ of its edges, is there a drawing of $G$ in which all edges of $F$ are free of crossings? We show that this question can be solved in polynomial time using a Hanani-Tutte style approach. If we require the drawing of $G$ to be straight-line, and allow at most one crossing along each edge in $F$, the problem turns out to be as hard as the existential theory of the real numbers.
Simple drawings of graphs are those in which each pair of edges share at most one point, either a common endpoint or a proper crossing. In this paper we study the problem of extending a simple drawing $D(G)$ of a graph $G$ by inserting a set of edges from the complement of $G$ into $D(G)$ such that the result is a simple drawing. In the context of rectilinear drawings, the problem is trivial. For pseudolinear drawings, the existence of such an extension follows from Levi's enlargement lemma. In contrast, we prove that deciding if a given set of edges can be inserted into a simple drawing is NP…
We investigate the problem of drawing two posets of the same ground set so that one is drawn from left to right and the other one is drawn from the bottom up. The input to this problem is a directed graph $G = (V, E)$ and two sets $X, Y$ with $X \cup Y = E$, each of which can be interpreted as a partial order of $V$. The task is to find a planar drawing of $G$ such that each directed edge in $X$ is drawn as an $x$-monotone edge, and each directed edge in $Y$ is drawn as a $y$-monotone edge. Such a drawing is called an $xy$-planar drawing. Testing whether a graph admits an $xy$-planar drawing i…
Let $G$ be a simple topological graph and let $Γ$ be a polyline drawing of $G$. We say that $Γ$ \emph{partially preserves the topology} of $G$ if it has the same external boundary, the same rotation system, and the same set of crossings as $G$. Drawing $Γ$ fully preserves the topology of $G$ if the planarization of $G$ and the planarization of $Γ$ have the same planar embedding. We show that if the set of crossing-free edges of $G$ forms a connected spanning subgraph, then $G$ admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at …
We study the problem of selecting a maximum-weight subgraph of a given graph such that the subgraph can be drawn within a prescribed drawing area subject to given non-uniform vertex sizes. We develop and analyze heuristics both for the general (undirected) case and for the use case of (directed) calculation graphs which are used to analyze the typical mistakes that high school students make when transforming mathematical expressions in the process of calculating, for example, sums of fractions.
The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar connected graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be \NP-hard, it is natural to consider the case where only a smal…
We introduce and study level-planar straight-line drawings with a fixed number $λ$ of slopes. For proper level graphs, we give an $O(n \log^2 n / \log \log n)$-time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present $O(n^{4/3} \log n)$-time and $O(λ n^…
We study the planar orthogonal drawing style within the framework of partial representation extension. Let $(G,H,Γ_H )$ be a partial orthogonal drawing, i.e., G is a graph, $H\subseteq G$ is a subgraph and $Γ_H$ is a planar orthogonal drawing of H. We show that the existence of an orthogonal drawing $Γ_G$ of $G$ that extends $Γ_H$ can be tested in linear time. If such a drawing exists, then there also is one that uses $O(|V(H)|)$ bends per edge. On the other hand, we show that it is NP-complete to find an extension that minimizes the number of bends or has a fixed number of bends per edge.
We study the problem of drawing a dynamic graph, where each vertex appears in the graph at a certain time and remains in the graph for a fixed amount of time, called the window size. This defines a graph story, i.e., a sequence of subgraphs, each induced by the vertices that are in the graph at the same time. The drawing of a graph story is a sequence of drawings of such subgraphs. To support readability, we require that each drawing is straight-line and planar and that each vertex maintains its placement in all the drawings. Ideally, the area of the drawing of each subgraph should be a functi…
The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph $G$, a connected subgraph $H$ of $G$ and a drawing $\mathcal{H}$ of $H$, the extension problem asks whether $\mathcal{H}$ can be extended into a drawing of $G$ while maintaining some desired property of the drawing (e.g., planarity). In their breakthrough result, Angelini et al. [ACM TALG 2015] showed that the extension problem is polynomial-time solvable when the aim is to preserve planarity. Very recently we considered this pr…
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 are given a tuple $(G,H,\mathcal{H})$ consisting of a graph $G$, a connected subgraph $H$ of $G$ and a drawing $\mathcal{H}$ of $H$, and the task is to extend $\mathcal{H}$ into a drawing of $G$ while maintaining some desired property of the drawing, such as planarity. In this paper we study the problem of extending partial 1-planar drawings, which are drawi…
We are developing a rule-based implementation of a tool to analyse and generate graphs. It is currently used in the domain of mason's marks. For thousands of years, stonemasons have been inscribing these symbolic signs on dressed stone. Geometrically, mason's marks are line drawings. They consist of a pattern of straight lines, sometimes circles and arcs. We represent mason's marks by connected planar graphs. Our prototype tool for analysis and generation of graphs is written in the rule-based declarative language Constraint Handling Rules. It features - a vertex-centric logical graph represen…
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University（应用数学系，数学物理学院，查理大学）；Computer Science Institute, Faculty of Mathematics and Physics, Charles University（计算机科学研究所，数学物理学院，查理大学）
Motivated by the successful application of geometry to proving the Harary-Hill Conjecture for "pseudolinear" drawings of $K_n$, we introduce "pseudospherical" drawings of graphs. A spherical drawing of a graph $G$ is a drawing in the unit sphere $\mathbb{S}^2$ in which the vertices of $G$ are represented as points -- no three on a great circle -- and the edges of $G$ are shortest-arcs in $\mathbb{S}^2$ connecting pairs of vertices. Such a drawing has three properties: (1) every edge $e$ is contained in a simple closed curve $γ_e$ such that the only vertices in $γ_e$ are the ends of $e$; (2) if…
In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes in input an upward planar drawing $Γ_H$ of a subgraph $H$ of a directed graph $G$ and asks whether $Γ_H$ can be extended to an upward planar drawing of $G$. Our study fits into the line of research on the extensibility of partial representations, which has recently become a mainstream in Graph Drawing. We show the following results. First, we prove that the Upward Planarity Extension problem is NP-complete, even if $G$ has a prescribed upward embedding, the vertex set of $H$ coincides wit…
We combine the idea of confluent drawings with Sugiyama style drawings, in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed style drawings. The basic idea is to cover a layered graph by complete bipartite subgraphs (bicliques), then replace bicliques with tree-like structures. The biclique cover problem is reduced to a special edge coloring problem and solved by heuristic coloring algorithms. Our method can be extended to obtain multi-depth confluent layered drawings.
TU Wien, Austria（维也纳工业大学）；BITS Pilani, K K Birla Goa Campus, India（印度理工学院比拉尼分校卡尔·K·比尔拉果阿校区）；Trier University, Germany（特里尔大学）；University of Perugia, Italy（佩鲁贾大学）
We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem---to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph---and the simultaneous planarity problem---to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Ou…
This paper studies questions about duality between crossings and non-crossings in graph drawings via the notions of thickness and antithickness. The "thickness" of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the edges can be partitioned into $k$ noncrossing subgraphs. The "antithickness" of a graph $G$ is the minimum integer $k$ such that in some drawing of $G$, the edges can be partitioned into $k$ thrackles, where a "thrackle" is a set of edges, each pair of which intersect exactly once. (Here edges with a common endvertex $v$ are considered to intersect at $v$.)…
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been studied extensively in the context of crossing number problems. We consider a natural class of simple topological drawings of complete bipartite graphs, in which we require that one side of the vertex set bipartition lies on the outer boundary of the drawing. We investigate the combinatorics of such drawings. For this purpose, we define combinatorial encodi…
We consider drawings of graphs that contain dense subgraphs. We introduce intersection-link representations for such graphs, in which each vertex $u$ is represented by a geometric object $R(u)$ and in which each edge $(u,v)$ is represented by the intersection between $R(u)$ and $R(v)$ if it belongs to a dense subgraph or by a curve connecting the boundaries of $R(u)$ and $R(v)$ otherwise. We study a notion of planarity, called Clique Planarity, for intersection-link representations of graphs in which the dense subgraphs are cliques.
Motivated by cognitive experiments providing evidence that large crossing-angles do not impair the readability of a graph drawing, RAC (Right Angle Crossing) drawings were introduced to address the problem of producing readable representations of non-planar graphs by supporting the optimal case in which all crossings form 90° angles. In this work, we make progress on the problem of finding RAC drawings of graphs of low degree. In this context, a long-standing open question asks whether all degree-3 graphs admit straight-line RAC drawings. This question has been positively answered for the Hami…
We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This strengthens an NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is…
Inspired by the increasingly popular research on extending partial graph drawings, we propose a new perspective on the traditional and arguably most important geometric graph parameter, the crossing number. Specifically, we define the partially predrawn crossing number to be the smallest number of crossings in any drawing of a graph, part of which is prescribed on the input (not counting the prescribed crossings). Our main result - an FPT-algorithm to compute the partially predrawn crossing number - combines advanced ideas from research on the classical crossing number and so called partial pl…
The thickness of a graph $G=(V,E)$ with $n$ vertices is the minimum number of planar subgraphs of $G$ whose union is $G$. A polyline drawing of $G$ in $\mathbb{R}^2$ is a drawing $Γ$ of $G$, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of $Γ$ is the maximum number of bends per edge in $Γ$, and the layer complexity of $Γ$ is the minimum integer $r$ such that the set of polygonal chains in $Γ$ can be partitioned into $r$ disjoint sets, where each set corresponds…
Charles University, Prague, Czech Republic（布拉格查理大学）；Institut für Informatik, Universität Würzburg, Germany（维尔茨堡大学计算机科学研究所）；Università degli Studi di Perugia, Perugia, Italy（佩鲁贾大学）；Technische Universität München, Germany（慕尼黑工业大学）
Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii $r>0$ where the set of segments forms a straight line drawing $G=(V,E)$ of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for $r\in [d_{\min},ηd_{\max}]$ and some constant $η$ where $d_{\max}$ and $d_{\m…
A metric space $\mathcal{T}$ is a \emph{real tree} if for any pair of points $x, y \in \mathcal{T}$ all topological embeddings $σ$ of the segment $[0,1]$ into $\mathcal{T}$, such that $σ(0)=x$ and $σ(1)=y$, have the same image (which is then a geodesic segment from $x$ to $y$). A \emph{signed graph} is a graph where each edge has a positive or negative sign. The \emph{Sitting Closer to Friends than Enemies} problem in trees has a signed graph $S$ as an input. The purpose is to determine if there exists an injective mapping (called \emph{valid distance drawing}) from $V(S)$ to the points of a r…
