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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

In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size $\frac{n}{3}+c$ (where the constant $c$ depends on the connectivity), and even better bounds hold for planar graphs with minimum degree 4 and 5. In this paper, we investigate similar matching-bounds for {\em 1-planar} graphs, i.e., graphs that can be drawn such that every edge has at most one crossing. We show that every 1-planar graph with minimum degree 3 has a matching of size at least $\frac{1}{7}n+\frac{12}{7}$, and this is tight for some graphs. We provide similar bounds for 1-planar graphs with minimum degree 4 and 5, while the case of minimum degree 6 and 7 remains open.

It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight., 4 pages

Consider a set $P$ of $n$ points on the boundary of an axis-aligned square $Q$. We study the boundary-anchored packing problem on $P$ in which the goal is to find a set of interior-disjoint axis-aligned rectangles in $Q$ such that each rectangle is anchored (has a corner at some point in $P$), each point in $P$ is used to anchor at most one rectangle, and the total area of the rectangles is maximized. Here, a rectangle is anchored at a point $p$ in $P$ if one of its corners coincides with $p$. In this paper, we show how to solve this problem in time linear in $n$, provided that the points of $P$ are given in sorted order along the boundary of $Q$. We also consider the problem for anchoring squares and give an $O(n^4)$-time algorithm when the points in $P$ lie on two opposite sides of $Q$., 18 pages, 11 figures

Following the seminal work of Erlebach and van Leeuwen in SODA 2008, we introduce the minimum ply covering problem. Given a set $P$ of points and a set $S$ of geometric objects, both in the plane, our goal is to find a subset $S'$ of $S$ that covers all points of $P$ while minimizing the maximum number of objects covering any point in the plane (not only points of $P$). For objects that are unit squares and unit disks, this problem is NP-hard and cannot be approximated by a ratio smaller than 2. We present 2-approximation algorithms for this problem with respect to unit squares and unit disks. Our algorithms run in polynomial time when the optimum objective value is bounded by a constant. Motivated by channel-assignment in wireless networks, we consider a variant of the problem where the selected unit disks must be 3-colorable, i.e., colored by three colors such that all disks of the same color are pairwise disjoint. We present a polynomial-time algorithm that achieves a 2-approximate solution, i.e., a solution that is 6-colorable. We also study the weighted version of the problem in dimension one, where $P$ and $S$ are points and weighted intervals on a line, respectively. We present an algorithm that solves this problem in $O(n + m + M )$-time where $n$ is the number of points, $m$ is the number of intervals, and $M$ is the number of pairs of overlapping intervals. This repairs a solution claimed by Nandy, Pandit, and Roy in CCCG 2017., 15 pages

A sliding k-transmitter inside an orthogonal polygon P, for a fixed $$k\ge 0$$ , is a point guard that travels along an axis-parallel line segment s in P. The sliding k-transmitter can see a point $$p\in P$$ if the perpendicular from p onto s intersects the boundary of P in at most k points. In the Minimum Sliding k-Transmitters ( $${\hbox {ST}}_k$$ ) problem, the objective is to guard P with the minimum number of sliding k-transmitters. In this paper, we give a constant-factor approximation algorithm for the $${\hbox {ST}}_k$$ problem on P for any fixed $$k\ge 0$$ . Moreover, we show that the $${\hbox {ST}}_0$$ problem is NP-hard on orthogonal polygons with holes even if only horizontal sliding 0-transmitters are allowed. For $$k>0$$ , the problem is NP-hard even in the extremely restricted case where P is simple and monotone. Finally, we study art gallery theorems; i.e., we give upper and lower bounds on the number of sliding transmitters required to guard P relative to the number of vertices of P.

\(\varTheta _6\)-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.