Planar Drawing Research Articles

Planar drawings of fixed-mobile bigraphs

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other partition, together with the edges, must be added to the drawing without any restriction on their positions. We assume that G is planar and study the problem of finding a planar straight-line drawing of G. We show that deciding whether G admits such a drawing is NP-hard in the general case. Under the assumption that each mobile vertex is placed in the convex hull of its neighbors, we are able to prove that the problem is also in NP. Moreover, if the intersection graph of these convex hulls is a path, a cycle or, more generally, a cactus, the problem is polynomial-time solvable through a dynamic programming approach. Finally, we describe linear-time testing algorithms when the fixed vertices are collinear or when they lie on a finite set of horizontal levels (lines) and no edge can intersect a level except at its fixed vertex.

Universal Slope Sets for 1-Bend Planar Drawings

We prove that every set of $$\varDelta -1$$ slopes is 1-bend universal for the planar graphs with maximum vertex degree $$\varDelta $$ . This means that any planar graph with maximum degree $$\varDelta $$ admits a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges can be chosen in any given set of $$\varDelta -1$$ slopes. Our result improves over previous literature in three ways: Firstly, it improves the known upper bound of $$\frac{3}{2} (\varDelta -1)$$ on the 1-bend planar slope number; secondly, the previously known algorithms compute 1-bend planar drawings by using sets of $$O(\varDelta )$$ slopes that may vary depending on the input graph; thirdly, while these algorithms typically minimize the slopes at the expenses of constructing drawings with poor angular resolution, we can compute drawings whose angular resolution is at least $$\frac{\pi }{\varDelta -1}$$ , which is worst-case optimal up to a factor of $$\frac{3}{4}$$ . Our proofs are constructive and give rise to a linear-time drawing algorithm.

Авангардные детские журналы и газеты 1920–1930-х годов в Сибири

The purpose of this article is to analyze the magazines and newspapers for children and youth issued on the territory of Siberia in 1920s – 1930s. A great many children’s books were issued that years, moreover, the approach to design of that books and to the contents of writings for children changed significantly: the topics had to be actual, associated with the construction of the new society. At the same time, exactly in children’s press in 1920s, the new principles of book graphics were formed. There are a large number of magazines and newspapers aimed at youth audiences were published in Siberia in the 1920s and 1930s, but they did not have a long history. Some of them appeared only once or twice, after that they closed. But all the more interesting is the study of these rare publications as experiments that influenced how the Soviet children’s and youth magazine was formed. Viewing magazines and newspapers allows you to observe how the rubrication and the genre system of Soviet publications for children evolved, as well as identify trends that have become a definite “sign of the times”. The article explores archive materials and examines the contents of printed issues, peculiarities of the approaches to the inner composition of the material and design techniques, discovers the features of the “Soviet avant-garde” development in children’s and youth periodicals. It indicates that the majority of the Siberian Children’s and youth magazines issued within that period has demonstrated a strongly demonstrated ideological overtone, claiming its purpose raising the new type of human and orientation on the “iterature of fact”. The article covers the peculiarities of the illustration techniques in Siberian post-revolutionary magazines. The article marks that up to the mid – late 1920s, the children’s and youth periodicals design became composed of such elements as insets, plane drawings based on a contrast combination of black and white, photography and photographic compilation. Furthermore, it describes a number of self-presentation techniques, developed exactly by the avant-garde art. As can be seen from the above, it can be stated that Siberian children’s and youth journalism acquired the avant-garde trends of the first third of the 20th century, however, they haven’t been gradually and fully realized.

A Graphical Calculus for Semi-Groupal Categories

Around the year 1988, Joyal and Street established a graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics. For a deeper understanding of their work, we consider a similar graphical calculus for semi-groupal categories. We introduce two frameworks to formalize this graphical calculus, a topological one based on the notion of a processive plane graph and a combinatorial one based on the notion of a planarly ordered processive graph, which serves as a combinatorial counterpart of a deformation class of processive plane graphs. We demonstrate the equivalence of Joyal and Street's graphical calculus and the theory of upward planar drawings. We introduce the category of semi-tensor schemes, and give a construction of a free monoidal category on a semi-tensor scheme. We deduce the unit convention as a kind of quotient construction, and show an idea to generalize the unit convention. Finally, we clarify the relation of the unit convention and Joyal and Street's construction of a free monoidal category on a tensor scheme.

HV-planarity: Algorithms and complexity

An HV-graph is a planar graph with vertex-degree at most four such that each edge is labeled either H (horizontal) or V (vertical). The HV-planarity testing problem asks whether an HV-graph admits an HV-drawing, that is, a planar drawing such that each edge with label H is drawn as a horizontal segment and each edge with label V is drawn as a vertical segment. We prove that the HV-planarity testing problem is NP-complete even for graphs with vertex-degree at most three, which answers an open question posed by both Manuch et al. [30] and Durocher et al. [17]. On the positive side, we prove that the HV-planarity testing problem can be solved in polynomial-time for series-parallel graphs. This result significantly extends a previous result by Durocher et al. about HV-planarity testing of biconnected outerplanar graphs of maximum vertex-degree three.

Windrose Planarity

Given a planar graph G and a partition of the neighbors of each vertex v in four sets v ↗ , v ↖ , v ↙ , and v ↘ , the problem W indrose P lanarity asks to decide whether G admits a windrose-planar drawing , that is, a planar drawing in which (i) each neighbor u ∈ v ↗ v is above and to the right of v , (ii) each neighbor u ∈ v ↖ is above and to the left of v , (iii) each neighbor u ∈ v ↙ is below and to the left of v , (iv) each neighbor u ∈ v ↘ is below and to the right of v , and (v) edges are represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow us to simultaneously visualize two partial orders defined by means of the edges of the graph. Although the problem is NP -hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a given combinatorial embedding. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph with n vertices that has a windrose-planar drawing, we can construct one with at most one bend per edge and with at most 2 n −5 bends in total, which lies on the 3 n × 3 n grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.

Drawing planar graphs with prescribed face areas

We study drawings of planar graphs where every inner face has a prescribed area. A plane graph is 'area-universal' if for every area assignment on the inner faces, there exists a straight-line drawing realizing the assigned areas. The only non-area-universal graph known so far is the octahedron graph. We give a simple counting argument that allows to prove non-area-universality for a large class of triangulations, namely Eulerian triangulations. Adding some geometric arguments, the concept allows to prove non-area universality for other graphs including the icosahedron. Relaxing the straight-line property by allowing the edges to bend, we show that one bend per edge is sufficient to realize any face area assignment of every plane graph. For plane bipartite graphs, it even suffices that half of the edges have a bend.

Morphing Schnyder Drawings of Planar Triangulations

We consider the problem of morphing between two planar drawings of the same triangulated graph, maintaining straight-line planarity. In “How to morph planar graph drawings” (SIAM Journal on Computing) the authors give a morph that consists of O(n) steps where each step is a linear morph that moves each of the n vertices in a straight line at uniform speed. However, their method imitates edge contractions so the grid size of the intermediate drawings is not bounded and the morphs are not good for visualization purposes. Using Schnyder embeddings, we are able to morph in $$O(n^2)$$ linear morphing steps and improve the grid size to $$O(n)\times O(n)$$ for a significant class of drawings of triangulations, namely the class of weighted Schnyder drawings. The morphs are visually attractive. Our method involves implementing the basic “flip” operations of Schnyder woods as linear morphs.

The Complexity of Computing the Cylindrical and the $t$-Circle Crossing Number of a Graph

A plane drawing of a graph is cylindrical if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The cylindrical crossing number of a graph \(G\) is the minimum number of crossings in a cylindrical drawing of \(G\). In his influential survey on the variants of the definition of the crossing number of a graph, Schaefer lists the complexity of computing the cylindrical crossing number of a graph as an open question. In this paper, we prove that the problem of deciding whether a given graph admits a cylindrical embedding is NP-complete, and as a consequence we show that the \(t\)-cylindrical crossing number problem is also NP-complete. Moreover, we show an analogous result for the natural generalization of the cylindrical crossing number, namely the \(t\)-crossing number.

Avant-garde Magazines for Children in Siberia, 1920–1930 (Omsk, Novosibirsk, Irkutsk) in Siberia

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Drawing planar graphs with many collinear vertices

Consider the following problem: Given a planar graph $G$, what is the maximum number $p$ such that $G$ has a planar straight-line drawing with $p$ collinear vertices? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known for it: Every $n$-vertex planar graph has a planar straight-line drawing with $\Omega(\sqrt{n})$ collinear vertices; for every $n$, there is an $n$-vertex planar graph whose every planar straight-line drawing has $O(n^\sigma)$ collinear vertices, where $\sigma<0.986$; every $n$-vertex planar graph of treewidth at most two has a planar straight-line drawing with $\Theta(n)$ collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs. This (partially) answers two open problems posed by Ravsky and Verbitsky [WG 2011:295 – 306]. Similar results are not possible for all bounded-treewidth planar graphs or for all bounded-degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.

Construction of a Topological Drawing of the Most Planar Subgraph of the Non-planar Graph

An algorithm was presented to construct a flat drawing of the non-planar graph. The source for solution of the problem is a set of isometric cycles of the graph, which allows one to reduce the solution to the discrete optimization methods. Consideration was given to the necessary concepts and structures for solution of the problem of constructing a planar topological graph drawing.

Every Plane Graph is Facially-Non-Repetitively $C$-choosable

A sequence $\left(x_1,x_2,\ldots,x_{2n}\right)$ of even length is a repetition if $\left(x_1,\ldots,x_n\right) =\left(x_{n+1},\ldots,x_{2n}\right)$. We prove existence of a constant $C &lt; 10^{4 \cdot 10^7}$ such that given any planar drawing of a graph $G$, and a list $L(v)$ of $C$ permissible colors for each vertex $v$ in $G$, there is a choice of a permissible color for each vertex such that the sequence of colors of the vertices on any facial simple path in $G$ is not a repetition.

Parameterized Complexity of 1-Planarity

We consider the problem of drawing graphs with at most one crossing per edge. These drawings, and the graphs that can be drawn in this way, are called $1$-planar. Finding $1$-planar drawings is known to be ${\mathsf{NP}}$-hard, but we prove that it is fixed-parameter tractable with respect to the vertex cover number, tree-depth, and cyclomatic number. Special cases of these algorithms provide polynomial-time recognition algorithms for $1$-planar split graphs and $1$-planar cographs. However, recognizing $1$-planar graphs remains ${\mathsf{NP}}$-complete for graphs of bounded bandwidth, pathwidth, or treewidth.

Drawing Planar Graphs with Few Geometric Primitives

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw a path with an arbitrary number of edges). Let $n$ denote the number of vertices of a graph. We show that trees can be drawn with $3n/4$ straight-line segments on a polynomial grid, and with $n/2$ straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with $(8n-17)/3$ segments on an $O(n)\times O(n^2)$ grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with $3n/2$ edges on an $O(n)\times O(n^2)$ grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only $(5n - 11)/3$ arcs. This is significantly smaller than the lower bound of $2n$ for line segments for a nontrivial graph class.

Non-aligned Drawings of Planar Graphs

A non-aligned drawing of a graph is a drawing where no two vertices are in the same row or column. Auber et al. showed that not all planar graphs have a non-aligned planar straight-line drawing in the $n\times n$-grid. They also showed that such a drawing exists if up to $n-3$ edges may have a bend. In this paper, we give algorithms for non-aligned planar drawings that improve on the results by Auber et al. In particular, we give such drawings on an $n\times n$-grid with at most $\frac{2n-5}{3}$ bends, and we study what grid-size can be achieved if we insist on having straight-line drawings.

Unified Hanani–Tutte Theorem

We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph $G$ has a drawing $D$ in the plane where every pair of independent edges crosses an even number of times, then $G$ has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in $D$. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.

On RAC drawings of 1-planar graphs

A drawing of a graph is 1-planar if each edge is crossed at most once. A graph is 1-planar if it has a 1-planar drawing. A k-bend RAC (Right Angle Crossing) drawing of a graph is a polyline drawing where each edge has at most k bends and edges cross only at right angles. A graph is k-bend RAC if it has a k-bend RAC drawing. A 0-bend RAC graph (drawing) is also called a straight-line RAC graph (drawing). The relationships between 1-planar and k-bend RAC graphs have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC and straight-line RAC graphs that are not 1-planar. The existence of 1-planar straight-line RAC drawings has been proven only for restricted families of 1-planar graphs. Two of the main questions still open are: (i) What is the complexity of deciding whether a graph has a drawing that is both 1-planar and straight-line RAC? (ii) Does every 1-planar graph have a drawing that is both 1-planar and 1-bend RAC? In this paper we answer these two questions. Namely, we prove an NP-hardness result for the first question, and we positively answer the second question by describing a drawing algorithm for 1-planar graphs.

Research on AutoCAD secondary development and function expansion based on VBA technology

AutoCAD is the most widely used drawing tool among the similar design drawing products. In the process of drawing different types of design drawings of the same product, there are a lot of repetitive and single work contents. The traditional manual method uses a drawing software AutoCAD drawing graphics with low efficiency, high error rate and high input cost shortcomings and many more. In order to solve these problems, the design of the parametric drawing system of the hot-rolled I-beam (steel beam) cross-section is completed by using the VBA secondary development tool and the Access database software with large-capacity storage data, and the analysis of the functional extension of the plane drawing and the parametric drawing design in this paper. For the secondary development of AutoCAD functions, the system drawing work will be simplified and work efficiency also has been greatly improved. This introduction of parametric design of AutoCAD drawing system to promote the industrial mass production and related industries economic growth rate similar to the standard I-beam hot-rolled products.

Simultaneous Embedding: Edge Orderings, Relative Positions, Cutvertices

A simultaneous embedding (with fixed edges) of two graphs $$G^{\textcircled {1}}$$ and $$G^{\textcircled {2}}$$ with common graph $$G=G^{\textcircled {1}} \cap G^{\textcircled {2}}$$ is a pair of planar drawings of $$G^{\textcircled {1}}$$ and $$G^{\textcircled {2}}$$ that coincide on G. It is an open question whether there is a polynomial-time algorithm that decides whether two graphs admit a simultaneous embedding (problem Sefe). In this paper, we present two results. First, a set of three linear-time preprocessing algorithms that remove certain substructures from a given Sefe instance, producing a set of equivalent Sefe instances without such substructures. The structures we can remove are (1) cutvertices of the union graph $$G^\cup = G^{\textcircled {1}} \cup G^{\textcircled {2}}$$ , (2) most separating pairs of $$G^\cup $$ , and (3) connected components of G that are biconnected but not a cycle. Second, we give an $$O(n^3)$$ -time algorithm solving Sefe for instances with the following restriction. Let u be a pole of a P-node $$\mu $$ in the SPQR-tree of a block of $$G^{\textcircled {1}}$$ or $$G^{\textcircled {2}}$$ . Then at most three virtual edges of $$\mu $$ may contain common edges incident to u. All algorithms extend to the sunflower case, i.e., to the case of more than two graphs pairwise intersecting in the same common graph.