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There exist many variants of guarding an orthogonal polygon in an orthogonal fashion: sometimes a guard can see within a rectangle, along a staircase, or along an orthogonal path with at most k bends. In this paper, we study all these guarding models for the special case of orthogonal polygons that have bounded treewidth in some sense. As our main result, we show that the problem of finding the minimum number of guards in all these models becomes linear-time solvable on polygons with bounded treewidth. We complement our main result by giving some hardness results.

A wheel graph consists of a cycle along with a center vertex connected to every vertex in the cycle. In this paper we show that every subgraph of a wheel graph has list coupled chromatic number at most 5, and this coloring can be found in linear time. We further show that ‘5’ is tight for every wheel graph with at least 5 vertices, and briefly discuss possible generalizations to planar graphs of treewidth 3.

We introduce a class of graphs called OAT graphs that generalizes \(P_4\)-sparse, chordal bipartite, and compact graphs. We prove that if G is a k-colourable OAT graph then G is \((k+1)\)-mixing and the \((k+1)\)-recolouring diameter of G is \(O(n^2)\), unifying and extending several results in the literature. We also identify a mistake in the literature and leave as an open problem whether a k-colourable \(P_5\)-free graph is \((k+1)\)-mixing.

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence---increasing or decreasing---has length at least three. By translating this sequence to a set of points ...

An upward drawing of a rooted tree is a drawing such that no parents are below their children. It is well-known that every rooted tree has a planar straight-line upward drawing of width at most log 2 ⁡ ( n + 1 ) . This is class-optimal, i.e., there are rooted trees that require log 2 ⁡ ( n + 1 ) width, but not instance-optimal, i.e., for some trees the drawing uses much more width than required. In this paper, we study how to find upward drawings with instance-optimal width. It turns out that their width is the same as the so-called Horton-Strahler number, which in turn is equivalent to the so-called rooted pathwidth. We give other properties of these graph parameters as well.

A (weak) rectangle visibility representation, or simply an RVR, of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Given a graph with a fixed embedding in the plane, we show that the problem of testing whether this graph has an embedding-preserving RVR can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs, i.e., for embedded graphs having at most one crossing per edge. The linear time algorithm uses three forbidden configurations, which extend the set known for straight-line drawings of 1-plane graphs. The algorithm first checks for the presence of these forbidden configurations in the input graph, and then either an embedding-preserving RVR is computed (also in linear time) or a forbidden configuration is reported as a negative witness. Finally, we discuss extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.

Special issue of selected papers from the 26th international symposium on graph drawing and network visualization (GD 2018) : guest editors' foreword

\(\varTheta _6\)-graphs are important geometric graphs that have many applications especially in wireless sensor networks. They are equivalent to Delaunay graphs where empty equilateral triangles take the place of empty circles. We investigate lower bounds on the size of maximum matchings in these graphs. The best known lower bound is n/3, where n is the number of vertices of the graph, which comes from half-\(\varTheta _6\)-graphs that are subgraphs of \(\varTheta _6\)-graphs. Babu et al. (2014) conjectured that any \(\varTheta _6\)-graph has a (near-)perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to \((3n-8)/7\).

Bobbin lace is a fibre art form in which threads are braided together to form a fabric, often with a very detailed and complex design. In traditional practice, each region of the fabric is filled with a periodic texture. We establish the groundwork for non-periodic lace patterns and present three new quasiperiodic families based on Sturmian words, the Penrose tiling by thick and thin rhombs and the Ammann-bar decoration of the Penrose tiling., Comment: 27 pages, 13 figures

It is well-known that both the pathwidth and the outer-planarity of a graph can be used to obtain lower bounds on the height of a planar straight-line drawing of a graph. But both bounds fall short for some graphs. In this paper, we consider two other parameters, the (simple) homotopy height and the (simple) grid-major height. We discuss the relationship between them and to the other parameters, and argue that they give lower bounds on the straight-line drawing height that are never worse than the ones obtained from pathwidth and outer-planarity., Comment: 28 pages, 11 figures. Expanded version of a paper in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019); for the proceedings version, see version 1

A measure for the visual complexity of a straight-line crossing-free drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph G, the minimum such number (over all drawings in dimension \(d \in \{2,3\}\)) is called the d-dimensional weak line cover number and denoted by \(\pi ^1_d(G)\). In 3D, the minimum number of planes needed to cover all vertices of G is denoted by \(\pi ^2_3(G)\). When edges are also required to be covered, the corresponding numbers \(\rho ^1_d(G)\) and \(\rho ^2_3(G)\) are called the (strong) line cover number and the (strong) plane cover number.

We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights.

Let T be an n-node tree of maximum degree 4, and let P be a set of n points in the plane with no two points on the same horizontal or vertical line. It is an open question whether T always has a planar drawing on P such that each edge is drawn as an orthogonal path with one bend (an “L-shaped” edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set P for which such drawings are possible to: \(O(n^{1.55})\) for maximum degree 4 trees; \(O(n^{1.22})\) for maximum degree 3 (binary) trees; and \(O(n^{1.142})\) for perfect binary trees.

In this paper, we study a class of graph drawings that arise from bobbin lace patterns. The drawings are periodic and require a combinatorial embedding with specific properties which we outline and demonstrate can be verified in linear time. In addition, a lace graph drawing has a topological requirement: it contains a set of non-contractible directed cycles which must be homotopic to (1, 0), that is, when drawn on a torus, each cycle wraps once around the minor meridian axis and zero times around the major longitude axis. We provide an algorithm for finding the two fundamental cycles of a canonical rectangular schema in a supergraph that enforces this topological constraint. The polygonal schema is then used to produce a straight-line drawing of the lace graph inside a rectangular frame. We argue that such a polygonal schema always exists for combinatorial embeddings satisfying the conditions of bobbin lace patterns, and that we can therefore create a pattern, given a graph with a fixed combinatorial embedding of genus one.

Computing a minimum-area planar straight-line drawing of a graph is known to be NP-hard for planar graphs, even when restricted to outerplanar graphs. However, the complexity question is open for trees. Only a few hardness results are known for straight-line drawings of trees under various restrictions such as edge length or slope constraints. On the other hand, there exist polynomial-time algorithms for computing minimum-width (resp., minimum-height) upward drawings of trees, where the height (resp., width) is unbounded.

This book constitutes the refereed proceedings of the 26th International Symposium on Graph Drawing and Network Visualization, GD 2018, held in Barcelona, Spain, in September 2018. The 41 full pape ...

The problem of partitioning an orthogonal polyhedron into a minimum number of boxes was shown to be NP-hard in 1991, but no approximability result is known except for a 4-approximation algorithm for 3D-histograms. In this paper we broaden the understanding of the 3D-histogram partitioning problem. We prove that partitioning a 3D-histogram into a minimum number of boxes is NP-hard, even for histograms of height two. This settles an open question posed by Floderus et al. We then show the problem to be APX-hard for histograms of height four. On the positive side, we give polynomial-time algorithms to compute optimal or approximate box partitions for some restricted but interesting classes of polyhedra and 3D-histograms.

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an xy-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is NP-hard in 2D.

In an EPG-representation of a graph G, each vertex is represented by a path in the rectangular grid, and (v, w) is an edge in G if and only if the paths representing v an w share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters.

A k -outerplanar graph is a graph that can be drawn in the plane without crossing such that after k -fold removal of the vertices on the outer-face there are no vertices left. In this paper, we study how to triangulate a k -outerplanar graph while keeping its outerplanarity small. Specifically, we show that not all k -outerplanar graphs can be triangulated so that the result is k -outerplanar, but they can be triangulated so that the result is ( k + 1 ) -outerplanar.

A segment representation of a graph is an assignment of line segments in 2D to the vertices in such a way that two segments intersect if and only if the corresponding vertices are adjacent. Not all graphs have such segment representations, but they exist, for example, for all planar graphs. In this note, we study the resolution that can be achieved for segment representations, presuming the ends of segments must be on integer grid points. We show that any planar graph (and more generally, any graph that has a so-called L -representation) has a segment representation in a grid of width and height 4 n .

We give an algorithm to create orthogonal drawings of 3-connected 3-regular plane graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 corners per face.

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.

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.

In this paper, we study planar drawings of maximal outerplanar graphs with the objective of achieving small height. (We do not necessarily preserve a given planar embedding.) A recent paper gave an algorithm for such drawings that is within a factor of 4 of the optimum height. In this paper, we substantially improve the approximation factor to become 2. The main ingredient is to define a new parameter of outerplanar graphs (the umbrella depth, obtained by recursively splitting the graph into graphs called umbrellas). We argue that the height of any poly-line drawing must be at least the umbrella depth, and then devise an algorithm that achieves height at most twice the umbrella depth.

This paper considers 1-string representations of planar graphs that are order-preserving in the sense that the order of crossings along the curve representing vertex v is the same as the order of edges in the clockwise order around v in the planar embedding. We show that this does not exist for all planar graphs (not even for all planar 3-trees), but show existence for some subclasses of planar partial 3-trees. In particular, for outer-planar graphs it can be order-preserving and outer-string in the sense that all ends of strings are on the outside of the representation.

A \(B_2\)-VPG representation of a graph is an intersection representation that consists of orthogonal curves with at most 2 bends. In this paper, we show that the curves of such a representation can be partitioned into \(O(\log n)\) groups that represent outer-string graphs or \(O(\log ^3 n)\) groups that represent permutation graphs. This leads to better approximation algorithms for hereditary graph problems, such as independent set, clique and clique cover, on \(B_2\)-VPG graphs.

A sliding camera inside an orthogonal polygon P is a point guard that travels back and forth along an orthogonal line segment \(\gamma \) in P. The sliding camera g can see a point p in P if the perpendicular from p onto \(\gamma \) is inside P. In this paper, we give the first constant-factor approximation algorithm for the problem of guarding P with the minimum number of sliding cameras. Next, we show that the sliding guards problem is linear-time solvable if the (suitably defined) dual graph of the polygon has bounded treewidth. On the other hand, we show that the problem is NP-hard on orthogonal polygons with holes even if only horizontal cameras are allowed. Finally, we study art gallery theorems for sliding cameras, thus, give upper and lower bounds in terms of the number of sliding cameras needed relative to the number of vertices n.

The segment minimization problem consists of finding a smallest set of binary matrices (segments), where non-zero values in each row of each matrix are consecutive, each matrix is assigned a positive integer weight (a segment-value), and the weighted sum of the matrices corresponds to the given input intensity matrix. This problem has direct applications in intensity-modulated radiation therapy, an effective form of cancer treatment.We study here the special case when the largest value H in the intensity matrix is small. We show that for an intensity matrix with one row, this problem is fixed-parameter tractable (FPT) in H; our algorithm obtains a significant asymptotic speedup over the previous best FPT algorithm. We also show how to solve the full-matrix problem faster than all previously known algorithms. Finally, we address a closely related problem that deals with minimizing the number of segments subject to a minimum beam-on time, defined as the sum of the segment-values, and again improve the running time of previous algorithms. Our algorithms have running time O(mn) in the case that the matrix has only entries in {0,1,2}.

We explore the art gallery problem for the special case that the domain (gallery) P is an m-polyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P. In particular, we show that $\lfloor\frac{m+1}{3} \rfloor$ point guards are always sufficient and sometimes necessary to cover an m-polyomino, possibly with holes. When $m \leq\frac{3n}{4} - 4$, the sufficiency condition yields a strictly lower guard number than $\lfloor\frac{n}{4}\rfloor$, given by the art gallery theorem for orthogonal polygons.

We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by point-contacts or side-contacts between the corresponding polygons. Specifically, we consider proportional contact representations, where pre-specified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2-segment graphs, which include maximal outerplanar graphs and partial 2-trees.

In this paper we study the problem of reconstructing orthogonal polyhedra from a putative vertex set, i.e., we are given a set of points and want to find an orthogonal polyhedron for which this is the set of vertices. This is well-studied in 2D; we mostly focus on 3D, and on the case where the given set of points may be rotated beforehand. We obtain fast algorithms for reconstruction in the case where the answer must be orthogonally convex.

Silhouettes of polyhedra are an important primitive in application areas such as machine vision and computer graphics. In this paper, we study how to select view points of convex polyhedra such that the silhouette satisfies certain properties. Specifically, we give algorithms to find all projections of a convex polyhedron such that a given set of edges, faces and/or vertices appear on the silhouette.We present an algorithm to solve this problem in O(k2) time for k edges. For orthogonal projections, we give an improved algorithm that is fully adaptive in the number l of connected components formed by the edges, and has a time complexity of O(klogk+kl). We then generalize this algorithm to edges and/or faces appearing on the silhouette.

Stoker's theorem states that in a convex polyhedron, the dihedral angles and edge lengths determine the facial angles if the graph is fixed. In this paper, we study under what conditions Stoker's theorem holds for orthogonal polyhedra, obtaining uniqueness and a linear-time algorithm in some cases, and NP-hardness in others.

We study the problem of reconstructing convex polygons and convex polyhedra given the number of visible edges and visible faces in some orthogonal projections. In 2D, we find necessary and sufficient conditions for the existence of a feasible polygon of size N and give an algorithm to construct one, if it exists. When N is not known, we give an algorithm to find the maximum and minimum sizes of a feasible polygon. In 3D, when the directions are covered by a single plane we show that a feasible polyhedron can be constructed from a feasible polygon. We also give an algorithm to construct a feasible polyhedron when the directions are covered by two planes. Finally, we show that the problem becomes NP-hard when the directions are covered by three or more planes.

The segment minimization problem consists of representing an integer matrix as the sum of the fewest number of integer matrices each of which have the property that the non-zeroes in each row are consecutive. This has direct applications to an effective form of cancer treatment. Using several insights, we extend previous results to obtain constant-factor improvements in the approximation guarantees. We show that these improvements yield better performance by providing an experimental evaluation of all known approximation algorithms using both synthetic and real-world clinical data. Our algorithms are superior for 76% of instances and we argue for their utility alongside the heuristic approaches used in practice.

In this paper, we study small planar drawings of planar graphs. For arbitrary planar graphs, Θ(n 2) is the established upper and lower bound on the worst-case area. A long-standing open problem is to determine for what graphs a smaller area can be achieved. We show here that series-parallel graphs can be drawn in O(n 3/2) area, and outerplanar graphs can be drawn in O(nlog n) area, but 2-outerplanar graphs and planar graphs of proper pathwidth 3 require Ω(n 2) area. Our drawings are visibility representations, which can be converted to polyline drawings of asymptotically the same area.

We consider the problem of sorting a permutation using a network of data structures as introduced by Knuth and Tarjan. In general the model as considered previously was restricted to networks that are directed acyclic graphs (DAGs) of stacks and/or queues. In this paper we study the question of which are the smallest general graphs that can sort an arbitrary permutation and what is their efficiency. We show that certain two-node graphs can sort in time @Q(n^2) and no simpler graph can sort all permutations. We then show that certain three-node graphs sort in time @W(n^3^/^2), and that there exist graphs of k nodes which can sort in time @Q(nlog"kn), which is optimal.

We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π∗ which minimizes the number of crossings. In voting and social science theory this is known as the Kemeny optimal aggregation problem minimizing the Kendall-τ distance. This rank aggregation problem can be phrased as a one-sided two-layer crossing minimization problem for a series of bipartite graphs or for an edge coloured bipartite graph, where crossings are counted only for monochromatic edges. We contribute the max version of the crossing minimization problem, which attempts to minimize the discrimination against any permutation. As our results, we correct the construction from [C. Dwork, R. Kumar, M. Noar, D. Sivakumar, Rank aggregation methods for the Web, Proc. WWW10 (2001) 613–622] and prove the NP-hardness of the common crossing minimization problem for k=4 permutations. Then we establish a 2−2/k-approximation, improving the previous factor of 2. The max version is shown NP-hard for every k≥4, and there is a 2-approximation. Both approximations are optimal, if the common permutation is selected from the given ones. For two permutations crossing minimization is solved by inspecting the drawings, whereas it remains open for three permutations.

In this paper, we study how to draw trees so that they are planar, straight-line and respect a given order of edges around each node. We focus on minimizing the height, and show that we can always achieve a height of at most 2pw(T)+1, where pw(T) (the so-called pathwidth) is a known lower bound on the height. Hence we give an asymptotic 2-approximation algorithm. We also create a drawing whose height is at most 3pw(T ), but where the width can be bounded by the number of nodes. Finally we construct trees that require height 2pw(T)+1 in all planar order-preserving straight-line drawings.

Any simple planar graph can be triangulated, i.e., we can add edges to it, without adding multi-edges, such that the result is planar and all faces are triangles. In this paper, we study the problem of triangulating a planar graph without increasing the pathwidth by much. We show that if a planar graph has pathwidth k, then we can triangulate it so that the resulting graph has pathwidth Ok where the factors are 1, 8 and 16 for 3-connected, 2-connected and arbitrary graphs. With similar techniques, we also show that any outer-planar graph of pathwidth k can be turned into a maximal outer-planar graph of pathwidth at most $$4k+4$$. The previously best known result here was $$16k+15$$.

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.

The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth. In this paper, we for the first time show that crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth~3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an $O(n)\times O(n)$-grid to achieve such a drawing. Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3, and a $4w^3$-approximation for maximal graphs of pathwidth $w$. This is a constant approximation for bounded pathwidth graphs., Comment: Full version of paper that will appear at ISAAC'17

A flipturn on a polygon consists of reversing the order of edges inside a pocket of the polygon, without changing lengths or slopes. Any polygon with n edges must be convexified after at most (n ź 1)! flipturns. A recent paper showed that in fact it will be convex after at most n(nź3)/2 flipturns.We give here lower bounds.We construct a polygon such that if pockets are chosen in a bad way, at least (n ź 2)2/4 flipturns are needed to convexify the polygon. In another construction, (n ź1)2/8 flipturns are needed, regardless of the order in which pockets are chosen. All our bounds are adaptive to a pre-specified number of distinct slopes of the edges.

In this paper, we study the problem of whether a polyhedron can be obtained from a net by folding along the creases. We show that this problem can be solved in polynomial time if the dihedral angle at each crease is given, and it becomes NP-hard if these angles are unknown. We also study the case when the net has rigid faces that should not intersect during the folding process.

An orthogonal drawing of a graph is an embedding of the graph in the rectangular grid, with vertices represented by axis-aligned boxes, and edges represented by paths in the grid that only possibly intersect at common endpoints. In this paper we study three-dimensional orthogonal drawings and provide lower bounds for three scenarios: (1) drawings where the vertices have a bounded aspect ratio, (2) drawings where the surfaces of vertices are proportional to their degrees, and (3) drawings without any such restrictions. Then we show that these lower bounds are asymptotically optimal, by providing constructions that in all scenarios match the lower bounds within a constant factor.

An independent set of a graph is a set of vertices without edges between them. Every planar graph has an independent set of size at least (1/4)n and there are planar graphs for which any independent set has size at most (1/4)n. In this paper similar bounds are provided for the problem of bounded-degree independent set, i.e., an independent set where additionally all vertices have degree less than a pre-specified bound D. Our upper and lower bounds match (up to a small constant) for D ? 16.

The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on the edges of the tree from the propagation of the shrinking process. In this paper, we ask the reverse question: Given a tree with directed edges, can it be the straight skeleton of a polygon? And if so, can we find a suitable simple polygon? We answer these questions for all directed trees where the order of edges around each node is fixed.

It is well-known that every graph with maximum degree 4 has an orthogonal drawing with area at most $$\frac{49}{64}n^2+On\approx 0.76n^2$$. In this paper, we show that if the graph is 3-connected, then the area can be reduced even further to $$\frac{9}{16}n^2+On \approx 0.56n^2$$. The drawing uses the 3-canonical order for not necessarily planar 3-connected graphs, which is a special Mondshein sequence and can hence be computed in linear time. To our knowledge, this is the first application of a Mondshein sequence in graph drawing.