Planar 3-trees Research Articles

Computing the Number and Average Size of Connected Sets in Planar 3-Trees

Computing the Number and Average Size of Connected Sets in Planar 3-Trees

An Improved Upper Bound on the Queue Number of Planar Graphs

A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)-rainbows, that is, k+1 independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, which is defined as the minimum value of k for which a k-queue layout is feasible. Recently, Dujmović et al. [J. ACM, 67(4), 22:1–38, 2020] showed that the queue number of planar graphs is at most 49, thus settling in the positive a long-standing conjecture by Heath, Leighton and Rosenberg. To achieve this breakthrough result, their approach involves three different techniques: (1) an algorithm to obtain 2-queue layouts of outerplanar graphs, (2) an algorithm to obtain 5-queue layouts of planar 3-trees, and (3) a decomposition of a planar graph into so-called tripods. In this work, we push further each of these techniques to obtain the first non-trivial improvement of the upper bound on the queue number of planar graphs from 49 to 42 .

On mixed linear layouts of series-parallel graphs

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. In 2017, Pupyrev disproved this conjecture by demonstrating a planar partial 3-tree that does not admit a mixed 1-stack 1-queue layout.In this work, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs. We conclude this by means of a stronger result, stating that the conjecture does not hold, even in the more general union and local settings. In the former, crossings (nestings) are allowed in the stack (queue) as long as the involved edges belong to different connected components of the subgraph induced by its edges, while in the latter several stacks and queues may form the layout, but locally every vertex is incident to edges from only one stack and one queue.

Planar L-Drawings of Bimodal Graphs

In a planar L-drawing of a directed graph (digraph) each edge $e$ is represented as a polyline composed of a vertical segment starting at the tail of $e$ and a horizontal segment ending at the head of $e$. Distinct edges may overlap, but not cross. Our main focus is on bimodal graphs, i.e., digraphs admitting a planar embedding in which the incoming and outgoing edges around each vertex are contiguous. We show that every plane bimodal graph without 2-cycles admits a planar L-drawing. This includes the class of upward-plane graphs. Bimodal graphs with 2-cycles admit a planar L-drawing if the underlying undirected graph with merged 2-cycles is a planar 3-tree. Finally, outerplanar digraphs admit a planar L-drawing - although they do not always have a bimodal embedding - but not necessarily with an outerplanar embedding.

On Mixed Linear Layouts of Series-Parallel Graphs

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. Recently, Pupyrev disproved this conjectured by demonstrating a planar partial 3-tree that does not admit a 1-stack 1-queue layout. In this note, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs.

Queue Layouts of Planar 3-Trees

A queue layout of a graph G consists of a linear order of the vertices of G and a partition of the edges of G into queues, so that no two independent edges of the same queue are nested. The queue number of graph G is defined as the minimum number of queues required by any queue layout of G. In this paper, we continue the study of the queue number of planar 3-trees, which form a well-studied subclass of planar graphs. Prior to this work, it was known that the queue number of planar 3-trees is at most seven. In this work, we improve this upper bound to five. We also show that there exist planar 3-trees whose queue number is at least four. Notably, this is the first example of a planar graph with queue number greater than three.

Long paths and toughness of [formula omitted]-trees and chordal planar graphs

We show that every k-tree of toughness greater than k3 is Hamilton-connected for k≥3. (In particular, chordal planar graphs of toughness greater than 1 are Hamilton-connected.) This improves the result of Broersma et al. (2007) and generalizes the result of Böhme et al. (1999).On the other hand, we present graphs whose longest paths are short. Namely, we construct 1-tough chordal planar graphs and 1-tough planar 3-trees, and we show that the shortness exponent of the class is 0, at most log3022, respectively. Both improve the bound of Böhme et al. Furthermore, the construction provides k-trees (for k≥4) of toughness greater than 1.

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.

Formula omitted]-page and [formula omitted]-page drawings with bounded number of crossings per edge

A drawing of a graph such that the vertices are drawn as points along a line and each edge is a circular arc in one of the two half-planes defined by this line is called a 2-page drawing. If all edges are in the same half-plane, the drawing is called a 1-page drawing. We want to compute 1-page and 2-page drawings of planar graphs such that the number of crossings per edge does not depend on the number of vertices. We show that for any constant k, there exist planar graphs that require more than k crossings per edge in both 1-page and 2-page drawings. We then prove that if the vertex degree is bounded by Δ, every planar 3-tree has a 2-page drawing with a number of crossings per edge that only depends on Δ. Finally, we show a similar result for 1-page drawings of partial 2-trees.

Intersection graphs of L-shapes and segments in the plane

An L-shape is the union of a horizontal and a vertical segment with a common endpoint. These come in four rotations: ▪, ▪,▪ and ▪. A k-bend path is a simple path in the plane, whose direction changes k times from horizontal to vertical. If a graph admits an intersection representation in which every vertex is represented by an ▪, an ▪ or ▪, a k-bend path, or a segment, then this graph is called an {▪}-graph, {▪,▪}-graph, Bk-VPG-graph or SEG-graph, respectively. Motivated by a theorem of Middendorf and Pfeiffer (1992), stating that every {▪,▪}-graph is a SEG-graph, we investigate several known subclasses of SEG-graphs and show that they are {▪}-graphs, or Bk-VPG-graphs for some small constant k. We show that all planar 3-trees, all line graphs of planar graphs, and all full subdivisions of planar graphs are {▪}-graphs. Furthermore we show that complements of planar graphs are B17-VPG-graphs and complements of full subdivisions are B2-VPG-graphs. Here a full subdivision is a graph in which each edge is subdivided at least once.

On self-approaching and increasing-chord drawings of 3-connected planar graphs

An $st$-path in a drawing of a graph is self-approaching if during the traversal of the corresponding curve from $s$ to any point $t'$ on the curve the distance to $t'$ is non-increasing. A path has increasing chords if it is self-approaching in both directions. A drawing is self-approaching (increasing-chord) if any pair of vertices is connected by a self-approaching (increasing-chord) path. We study self-approaching and increasing-chord drawings of triangulations and 3-connected planar graphs. We show that in the Euclidean plane, triangulations admit increasing-chord drawings, and for planar 3-trees we can ensure planarity. We prove that strongly monotone (and thus increasing-chord) drawings of trees and binary cactuses require exponential resolution in the worst case, answering an open question by Kindermann et al. (GD 2014). Moreover, we provide a binary cactus that does not admit a self-approaching drawing. Finally, we show that 3-connected planar graphs admit increasing-chord drawings in the hyperbolic plane and characterize the trees that admit such drawings.

Drawing Outer 1-planar Graphs with Few Slopes

A graph is outer 1-planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1-planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3-trees. We show that an outer 1-planar graph G of bounded degree ∆ admits an outer 1-planar straight-line drawing that uses O(∆) different slopes, which generalizes a previous result by Knauer et al. about the outerplanar slope number of outerplanar graphs (Knauer, Micek, and Walczak. CGTA, 2014). We also show that O(∆2) slopes suffice to construct a crossing-free straight-line drawing of G; the best known upper bound on the planar slope number of planar partial 3-trees of bounded degree ∆ is O(∆5) as proved by Jelínek et al. (V. Jelínek, E. Jelínková, J. Kratochvíl, B. Lidický, M. Tesar, and T. Vyskocil. Graphs and Combinatorics, 2013).

Universal point sets for planar three-trees

For every n∈N, we present a set Sn of O(n3/2log⁡n) points in the plane such that every planar 3-tree with n vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of Sn. This is the first subquadratic upper bound on the cardinality of universal point sets for planar 3-trees, as well as for the class of 2-trees and serial parallel graphs.

Drawing Graphs with Few Arcs

Let G=(V,E) be a planar graph. An arrangement of circular arcs is called a composite arc-drawing of G, if its 1-skeleton is isomorphic to G. Similarly, a composite segment-drawing is described by an arrangement of straight-line segments. We ask for the smallest possible ground set of arcs/segments for a composite arc/segment-drawing. We present algorithms for constructing composite arc-drawings with a small ground set for trees, series-parallel graphs, planar 3-trees and general planar graphs. In the case where G is a tree, we also introduce an algorithm that realizes the vertices of the composite drawing on a O(n1.81) ×n grid. For each of the graph classes we provide a lower bound for the maximal size of the arrangement's ground set.

Antiferromagnetic Ising model in triangulations with applications to counting perfect matchings

In this work we give a lower bound for the groundstate degeneracy of the antiferromagnetic Ising model in the class of stack triangulations, also known as planar 3-trees. The geometric dual graphs of stack triangulations form a class, say C, of cubic bridgeless planar graphs, i.e. G∈C iff its geometric dual graph is a planar 3-tree. As a consequence, we show that every graph G∈C has at least 3⋅φ(|V(G)|+8)/30≥3⋅2(|V(G)|+8)/44 distinct perfect matchings, where φ is the golden ratio. Our bound improves (slightly) upon the 3⋅2(|V(G)|+12)/60 bound obtained by Cygan, Pilipczuk, and Škrekovski (2013) for the number of distinct perfect matchings also for graphs G∈C with at least 8 nodes.Our work builds on an alternative perspective relating the number of perfect matchings of cubic bridgeless planar graphs and the number of so called groundstates of the widely studied Ising model from statistical physics. With hindsight, key steps of our work can be rephrased in terms of standard graph theoretic concepts, without resorting to terminology from statistical physics. Throughout, we draw parallels between the terminology we rely on and some of the concepts introduced/developed independently elsewhere.

Outerplanar graph drawings with few slopes

We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that Δ−1 edge slopes suffice for every outerplanar graph with maximum degree Δ⩾4. This improves on the previous bound of O(Δ5), which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every Δ⩾4 there is an outerplanar graph with maximum degree Δ that requires at least Δ−1 distinct edge slopes in an outerplanar straight-line drawing.

Linear-Time Algorithms for Hole-free Rectilinear Proportional Contact Graph Representations

In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we first study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12-sided rectilinear polygons and takes O(nlogn) time. We describe a new algorithm that guarantees 10-sided rectilinear polygons and runs in O(n) time. We also describe a linear-time algorithm for proportional contact representation of planar 3-trees with 8-sided rectilinear polygons and show that this is optimal, as there exist planar 3-trees that require 8-sided polygons. We then show that a maximal outer-planar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outer-boundary is a rectangle and with 4 sides otherwise. Finally we study maximal series-parallel graphs. Here we show that O(1)-sided rectilinear polygons are not possible unless we allow holes, but 6-sided polygons can be achieved with arbitrarily small holes.

Universal Line-Sets for Drawing Planar 3-Trees

A set S of lines is universal for drawing planar graphs with n vertices if every planar graph G with n vertices can be drawn on S such that each vertex of G is drawn as a point on a line of S and each edge is drawn as a straight-line segment without any edge crossing. It is known that $\lfloor \frac{2(n-1)}{3}\rfloor$ parallel lines are universal for any planar graph with n vertices. In this paper we show that a set of $\lfloor \frac{n-3}{2} \rfloor +3 $ parallel lines or a set of $\lceil \frac{n+3}{4} \rceil$ concentric circles are universal for drawing planar 3-trees with n vertices. In both cases we give linear-time algorithms to find such drawings. A by-product of our algorithm is the generalization of the known bijection between plane 3-trees and rooted full ternary trees to the bijection between planar 3-trees and unrooted full ternary trees. We also identify some subclasses of planar 3-trees whose drawings are supported by fewer than $\lfloor \frac{n-3}{2} \rfloor +3 $ parallel lines.

Drawing planar 3-trees with given face areas

We study straight-line drawings of planar graphs such that each interior face has a prescribed area. It was known that such drawings exist for all planar graphs with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees, i.e., subgraphs of a triangulated planar graph obtained by repeatedly inserting a vertex in one triangle and connecting it to all vertices of the triangle. Moreover, vertices have rational coordinates if the face areas are rational, and we can bound the resolution. We also give some negative results for other graph classes.

Point-set embeddings of plane 3-trees

A straight-line drawing of a plane graph G is a planar drawing of G, where each vertex is drawn as a point and each edge is drawn as a straight line segment. Given a set S of n points in the Euclidean plane, a point-set embedding of a plane graph G with n vertices on S is a straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. The problem of deciding if G admits a point-set embedding on S is NP-complete in general and even when G is 2-connected and 2-outerplanar. In this paper, we give an O ( n 2 ) time algorithm to decide whether a plane 3-tree admits a point-set embedding on a given set of points or not, and find an embedding if it exists. We prove an Ω ( n log n ) lower bound on the time complexity for finding a point-set embedding of a plane 3-tree. We then consider a variant of the problem, where we are given a plane 3-tree G with n vertices and a set S of k > n points, and present a dynamic programming algorithm to find a point-set embedding of G on S if it exists. Furthermore, we show that the point-set embeddability problem for planar partial 3-trees is also NP-complete.