Combinatorial Embedding Research Articles

On planarity of graphs in homotopy type theory

AbstractIn this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.

Unimodular random one-ended planar graphs are sofic

AbstractWe prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs.

How to Fit a Tree in a Box

We study compact straight-line embeddings of trees. We show that perfect binary trees can be embedded optimally: a tree with n nodes can be drawn on a \(\sqrt{n}\) by \(\sqrt{n}\) grid. We also show that testing whether a given rooted binary tree has an upward embedding with a given combinatorial embedding in a given grid is NP-hard.

The Density of Fan-Planar Graphs

A topological drawing of a graph is fan-planar if for each edge $e$ the edges crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its crossing edges. A fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by three forbidden patterns, one of which is the configuration where $e$ is crossed by two independent edges and the other two where $e$ is crossed by two incident edges in a way that encloses some endpoint of $e$. A topological drawing is simple if any two edges have at most one point in common.
 Fan-planar graphs are a new member in the ever-growing list of topological graphs defined by forbidden intersection patterns, such as planar graphs and their generalizations, Turán-graphs and Conway's thrackle conjecture. Hence fan-planar graphs fall into an important field in combinatorial geometry with applications in various areas of discrete mathematics.
 As every $1$-planar graph is fan-planar and every fan-planar graph is $3$-quasiplanar, they also fit perfectly in a recent series of works on nearly-planar graphs from the areas of graph drawing and combinatorial embeddings.
 In this paper we show that every fan-planar graph on $n$ vertices has at most $5n-10$ edges, even though a fan-planar drawing may have a quadratic number of crossings. Our bound, which is tight for every $n \geq 20$, indicates how nicely fan-planar graphs fit in the row with planar graphs ($3n-6$ edges) and $1$-planar graphs ($4n-8$ edges). With this, fan-planar graphs form an inclusion-wise largest non-trivial class of topological graphs defined by forbidden patterns, for which the maximum number of edges on $n$ vertices is known exactly.
 We demonstrate that maximum fan-planar graphs carry a rich structure, which makes this class attractive for many algorithms commonly used in graph drawing. Finally, we discuss possible extensions and generalizations of these new concepts.

Markov Models of Coarsening in Two-Dimensional Foams with Edge Rupture

We construct Markov processes for modeling the rupture of edges in a two-dimensional foam. We first describe a network model for tracking topological information of foam networks with a state space of combinatorial embeddings. Through a mean-field rule for randomly selecting neighboring cells of a rupturing edge, we consider a simplified version of the network model in the sequence space $\ell_1(\mathbb N)$ which counts total numbers of cells with $n\ge 3$ sides ($n$-gons). Under a large cell limit, we show that number densities of $n$-gons in the mean field model are solutions of an infinite system of nonlinear kinetic equations. This system is comparable to the Smoluchowski coagulation equation for coalescing particles under a multiplicative collision kernel, suggesting gelation behavior. Numerical simulations reveal gelation in the mean-field model, and also comparable statistical behavior between the network and mean-field models.

Radial Level Planarity with Fixed Embedding

We study level planarity testing of graphs with a fixed combinatorial embedding for three different notions of combinatorial embeddings, namely the level embedding, the upward embedding and the planar embedding. These notions allow for increasing degrees of freedom in their corresponding drawings. For the fixed level embedding there are known and easy to test level planarity criteria. We use these criteria to prove an "untangling" lemma that plays a key role in a simple level planarity test for the case where only the upward embedding is fixed. This test is then adapted to the case where only the planar embedding is fixed. Further, we characterize radial upward planar embeddings, which lets us extend our results to radial level planarity. No algorithms were previously known for these problems.

Tilings by hexagonal prisms and embeddings into primitive cubic networks.

All possible combinatorial embeddings into primitive cubic networks of arbitrary tilings of 3D space by pairwise congruent and parallel regular hexagonal prisms are discussed and classified.

Embedding Graphs into Embedded Graphs

A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing whether a piece-wise linear drawing of a planar graph G in the plane is approximable by an embedding can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class. In other words, we show that c-planarity with embedded pipes is tractable for graphs with prescribed combinatorial embedding. To the best of our knowledge, an analogous result was previously known essentially only when G is a cycle.

An effective crossing minimisation heuristic based on star insertion

We present a new heuristic method for minimising crossings in a graph. The method is based upon repeatedly solving the so-called star insertion problem in the setting where the combinatorial embedding is fixed, and has several desirable characteristics for practical use. We introduce the method, discuss some aspects of algorithm design for our implementation, and provide some experimental results. The results indicate that our method compares well to existing methods, and also that it is suitable for dense instances.

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.

C-planarity of embedded cyclic c-graphs

We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.

Combinatorial embedding of chain transitive zero-dimensional systems into chaos

In this paper, we construct examples of transitive Li–Yorke chaotic zero-dimensional topological dynamical systems, which should be able to accommodate of a variety of subsystems. In fact, we show that a zero-dimensional chain transitive dynamical system can be embedded into a transitive Li–Yorke chaotic system with a dense scrambled set, and with the same topological entropy. Our work reflects the notion of uniform chaos, which was recently developed. In addition, our work reflects the recent work of invariance of scrambled sets in general topological setting. In these works, Mycielski sets play an essential role. Actually, we concretely construct a Mycielski set K such that it is densely uniformly chaotic, and also invariant in both directions. Furthermore, every point in K is positively and negatively transitive. The uniform proximality and recurrence of K are also bidirectional.

Strip Planarity Testing for Embedded Planar Graphs

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 edge is represented by a curve that is monotone in the y-direction and, for any $$u,v\in V$$ with $$\gamma (u)<\gamma (v)$$ , it holds that $$y(u)<y(v)$$ . The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. Most notably, we provide a polynomial-time reduction from strip planarity testing to clustered planarity. We show that the strip planarity testing problem is polynomial-time solvable if G has a prescribed combinatorial embedding.

Drawing Partially Embedded and Simultaneously Planar Graphs

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, with a number of crossings between any pair of edges which is bounded by a constant. Our proofs provide efficient algorithms if the combinatorial embedding of the drawing is given. Our result on partially embedded graph drawing generalizes a classic result by Pach and Wenger which shows that any planar graph can be drawn with a linear number of bends per edge if the location of each vertex is fixed.

A new perspective on clustered planarity as a combinatorial embedding problem

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges.Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.

THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

Orthogonal Graph Drawing with Flexibility Constraints

Traditionally, the quality of orthogonal planar drawings is quantified by either the total number of bends, or the maximum number of bends per edge. However, this neglects that in typical applications, edges have varying importance. In this work, we investigate an approach that allows to specify the maximum number of bends for each edge individually, depending on its importance. We consider a new problem called FlexDraw that is defined as follows. Given a planar graph G=(V,E) on n vertices with maximum degree 4 and a function $\operatorname{flex}: E \longrightarrow\mathbb{N}_{0}$ that assigns a flexibility to each edge, does G admit a planar embedding on the grid such that each edge e has at most $\operatorname{flex}(e)$ bends? Note that in our setting the combinatorial embedding of G is not fixed. FlexDraw directly extends the problem β-embeddability asking whether G can be embedded with at most β bends per edge. We give an algorithm with running-time O(n 2) solving FlexDraw when the flexibility of each edge is positive. This includes 1-embeddability as a special case and thus closes the complexity gap between 0-embeddability, which is $\mathcal{NP}$ -hard to decide, and 2-embeddability, which is efficiently solvable since every planar graph with maximum degree 4 admits a 2-embedding except for the octahedron. In addition to the polynomial-time algorithm we show that FlexDraw is $\mathcal{NP}$ -hard even if the edges with flexibility 0 induce a tree or a union of disjoint stars.

Cracking Analysis of Spiral Case Structure with Combinatorial Embedding Manner of Large Underground Power Station Turbine Unit

The spiral case with combinatorial embedding manner is a novel design and inherits the advantage of the general types of embedding manners of embedded with cushion layers and embedded directly, which is progressively used in hydro projects. In the present paper, the mechanical behavior of a larger spiral case with combinatorial embedding manner is analyzed by using 3-D non-linear finite element method. The deformation of the spiral case structure and crack analysis of outer reinforced concrete structure are carried out. The strengthening measures under the engineering control criterion are also discussed. The results show that the penetrating crack will appear in the concrete and the relative rising displacement of generator pedestal structure is out of the control criterion if no strengthening measures implemented. Increasing the quantity of reinforcement only is not able to hold back the penetrating cracks. The enhancement of the tensile strength of concrete is an effective way to decrease the range of crack propagation, resist the penetrating and decrease the rising displacement of generator pedestal structure. It also indicated that extending the laying length of cushion is another effective way to decrease the rising displacement of generator pedestal structure

Generalized [formula omitted]-ary tanglegrams on level graphs: A satisfiability-based approach and its evaluation

A tanglegram is a pair of (not necessarily binary) trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in computational biology to compare evolutionary histories of species. In this work we present a formulation of two related combinatorial embedding problems concerning tanglegrams in terms of CNF-formulas. The first problem is known as the planar embedding and the second as the crossing minimization problem. We show that our satisfiability-based encoding of these problems can handle a much more general case with more than two, not necessarily binary or complete, trees defined on arbitrary sets of leaves and allowed to vary their layouts. Furthermore, we present an experimental comparison of our technique and several known heuristics for solving generalized binary tanglegrams, showing its competitive performance and efficiency and thus proving its practical usability.

The Straight-Line RAC Drawing Problem is NP-Hard

Recent cognitive experiments have shown that the negative impact of an edge crossing on the human understanding of a graph drawing, tends to be eliminated in the case where the crossing angles are greater than 70 degrees. This motivated the study of RAC drawings, in which every pair of crossing edges intersects at right angle. In this work, we demonstrate a class of graphs with unique RAC combinatorial embedding and we employ members of this class in order to show that it is NP-hard to decide whether a graph admits a straight-line RAC drawing.