Search

We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an $n$-vertex planar graph and two planar straight-line drawings of the graph on an $O(n) \times O(n)$ grid. The planarity-preserving morph is composed of $O(n)$ linear morphs between successive pairs of drawings, each on an $O(n) \times O(n)$ grid with a constant number of bends per edge. The algorithm to compute the morph runs in $O(n^2)$ time on a word RAM model with standard arithmetic operations -- in particular no square roots or cube roots are required. The first step of the algorithm is to morph each input drawing to a planar orthogonal box drawing where vertices are represented by boxes and each edge is drawn as a horizontal or vertical segment. The second step is to morph between planar orthogonal box drawings. This is done by extending known techniques for morphing planar orthogonal drawings with point vertices., Comment: To appear in the proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

A polyiamond is a polygon composed of unit equilateral triangles, and a generalized deltahedron is a convex polyhedron whose every face is a convex polyiamond. We study a variant where one face may be an exception. For a convex polygon P, if there is a convex polyhedron that has P as one face and all the other faces are convex polyiamonds, then we say that P can be domed. Our main result is a complete characterization of which equiangular n-gons can be domed: only if n is in {3, 4, 5, 6, 8, 10, 12}, and only with some conditions on the integer edge lengths., Comment: 25 pages, 19 figures, 8 references

We explore an Art Gallery variant where each point of a polygon must be seen by k guards, and guards cannot see through other guards. Surprisingly, even covering convex polygons under this variant is not straightforward. For example, covering every point in a triangle k=4 times (a 4-cover) requires 5 guards, and achieving a 10-cover requires 12 guards. Our main result is tight bounds on k-covering a convex polygon of n vertices, for all k and n. The proofs of both upper and lower bounds are nontrivial. We also obtain bounds for simple polygons, leaving tight bounds an open problem., Comment: 23 pages, 16 figures, 9 references

The geodesic edge center of a polygon is a point c inside the polygon that minimizes the maximum geodesic distance from c to any edge of the polygon, where geodesic distance is the shortest path distance inside the polygon. We give a linear-time algorithm to find a geodesic edge center of a simple polygon. This improves on the previous O(n log n) time algorithm by Lubiw and Naredla [European Symposium on Algorithms, 2021]. The algorithm builds on an algorithm to find the geodesic vertex center of a simple polygon due to Pollack, Sharir, and Rote [Discrete & Computational Geometry, 1989] and an improvement to linear time by Ahn, Barba, Bose, De Carufel, Korman, and Oh [Discrete & Computational Geometry, 2016]. The geodesic edge center can easily be found from the geodesic farthest-edge Voronoi diagram of the polygon. Finding that Voronoi diagram in linear time is an open question, although the geodesic nearest edge Voronoi diagram (the medial axis) can be found in linear time. As a first step of our geodesic edge center algorithm, we give a linear-time algorithm to find the geodesic farthest-edge Voronoi diagram restricted to the polygon boundary.

For a set $P$ of $n$ points in the plane in general position, a non-crossing spanning tree is a spanning tree of the points where every edge is a straight-line segment between a pair of points and no two edges intersect except at a common endpoint. We study the problem of reconfiguring one non-crossing spanning tree of $P$ to another using a sequence of flips where each flip removes one edge and adds one new edge so that the result is again a non-crossing spanning tree of $P$. There is a known upper bound of $2n-4$ flips [Avis and Fukuda, 1996] and a lower bound of $1.5n - 5$ flips. We give a reconfiguration algorithm that uses at most $2n-3$ flips but reduces that to $1.5n-2$ flips when one tree is a path and either: the points are in convex position; or the path is monotone in some direction. For points in convex position, we prove an upper bound of $2d - \Omega(\log d)$ where $d$ is half the size of the symmetric difference between the trees. We also examine whether the happy edges (those common to the initial and final trees) need to flip, and we find exact minimum flip distances for small point sets using exhaustive search., Comment: 25 pages

The flip graph for a set $P$ of points in the plane has a vertex for every triangulation of $P$, and an edge when two triangulations differ by one flip that replaces one triangulation edge by another. The flip graph is known to have some connectivity properties: (1) the flip graph is connected; (2) connectivity still holds when restricted to triangulations containing some constrained edges between the points; (3) for $P$ in general position of size $n$, the flip graph is $\lceil \frac{n}{2} -2 \rceil$-connected, a recent result of Wagner and Welzl (SODA 2020). We introduce the study of connectivity properties of the flip graph when some edges between points are forbidden. An edge $e$ between two points is a flip cut edge if eliminating triangulations containing $e$ results in a disconnected flip graph. More generally, a set $X$ of edges between points of $P$ is a flip cut set if eliminating all triangulations that contain edges of $X$ results in a disconnected flip graph. The flip cut number of $P$ is the minimum size of a flip cut set. We give a characterization of flip cut edges that leads to an $O(n \log n)$ time algorithm to test if an edge is a flip cut edge and, with that as preprocessing, an $O(n)$ time algorithm to test if two triangulations are in the same connected component of the flip graph. For a set of $n$ points in convex position (whose flip graph is the 1-skeleton of the associahedron) we prove that the flip cut number is $n-3$.

The input to the distant representatives problem is a set of $n$ objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are axis-aligned rectangles, we give polynomial time constant-factor approximation algorithms for the $L_1$, $L_2$, and $L_\infty$ distance measures. We also prove lower bounds on the approximation factors that can be achieved in polynomial time (unless P = NP)., Comment: Full-length version of a paper to appear at ESA'21

We introduce the \emph{visibility center} of a set of points inside a polygon -- a point $c_V$ such that the maximum geodesic distance from $c_V$ to see any point in the set is minimized. For a simple polygon of $n$ vertices and a set of $m$ points inside it, we give an $O((n+m) \log {(n+m)})$ time algorithm to find the visibility center. We find the visibility center of \emph{all} points in a simple polygon in $O(n \log n)$ time. Our algorithm reduces the visibility center problem to the problem of finding the geodesic center of a set of half-polygons inside a polygon, which is of independent interest. We give an $O((n+k) \log (n+k))$ time algorithm for this problem, where $k$ is the number of half-polygons., Comment: Full-length version of a paper that appeared at the European Symposium of Algorithms 2021

Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree. These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems); polynomial-time algorithms for simple graph classes such as cliques, stars, paths, and cycles; and constant-factor approximation algorithms in some cases. The two natural cases of sequential and parallel token swapping in trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown. We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is $2$) and show that no such algorithm can achieve an approximation factor less than $2$.

Let $P$ and $Q$ be finite point sets of the same cardinality in $\mathbb{R}^2$, each labelled from $1$ to $n$. Two noncrossing geometric graphs $G_P$ and $G_Q$ spanning $P$ and $Q$, respectively, are called compatible if for every face $f$ in $G_P$, there exists a corresponding face in $G_Q$ with the same clockwise ordering of the vertices on its boundary as in $f$. In particular, $G_P$ and $G_Q$ must be straight-line embeddings of the same connected $n$-vertex graph. Deciding whether two labelled point sets admit compatible geometric paths is known to be NP-complete. We give polynomial-time algorithms to find compatible paths or report that none exist in three scenarios: $O(n)$ time for points in convex position; $O(n^2)$ time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and $O(n^2 \log n)$ time for points in general position if the paths are restricted to be monotone., Comment: A preliminary version of the paper was presented at the 30th Canadian Conference on Computational Geometry (CCCG 2018)

Given two $n$-vertex polygons, $P=(p_1, \ldots, p_n)$ lying in the $xy$-plane at $z=0$, and $P'=(p'_1, \ldots, p'_n)$ lying in the $xy$-plane at $z=1$, a banded surface is a triangulated surface homeomorphic to an annulus connecting $P$ and $P'$ such that the triangulation's edge set contains vertex disjoint paths $\pi_i$ connecting $p_i$ to $p'_i$ for all $i =1, \ldots, n$. The surface then consists of bands, where the $i$th band goes between $\pi_i$ and $\pi_{i+1}$. We give a polynomial-time algorithm to find a banded surface without Steiner points if one exists. We explore connections between banded surfaces and linear morphs, where time in the morph corresponds to the $z$ direction. In particular, we show that if $P$ and $P'$ are convex and the linear morph from $P$ to $P'$ (which moves the $i$th vertex on a straight line from $p_i$ to $p'_i$) remains planar at all times, then there is a banded surface without Steiner points., Comment: preliminary version appeared in the Canadian Conference on Computational Geometry (CCCG) 2019

A $k$-colouring of a graph $G$ is an assignment of at most $k$ colours to the vertices of $G$ so that adjacent vertices are assigned different colours. The reconfiguration graph of the $k$-colourings, $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge in $\mathcal{R}_k(G)$ if they differ in colour on exactly one vertex. For a $k$-colourable graph $G$, we investigate the connectivity and diameter of $\mathcal{R}_{k+1}(G)$. It is known that not all weakly chordal graphs have the property that $\mathcal{R}_{k+1}(G)$ is connected. On the other hand, $\mathcal{R}_{k+1}(G)$ is connected and of diameter $O(n^2)$ for several subclasses of weakly chordal graphs such as chordal, chordal bipartite, and $P_4$-free graphs. We introduce a new class of graphs called OAT graphs that extends the latter classes and in fact extends outside the class of weakly chordal graphs. OAT graphs are built from four simple operations, disjoint union, join, and the addition of a clique or comparable vertex. We prove that if $G$ is a $k$-colourable OAT graph then $\mathcal{R}_{k+1}(G)$ is connected with diameter $O(n^2)$. Furthermore, we give polynomial time algorithms to recognize OAT graphs and to find a path between any two colourings in $\mathcal{R}_{k+1}(G)$., Comment: 24 pages, 8 figures

Given a flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on a region of the plane) with a mountain-valley (MV) assignment $\mu:E\to\{-1,1\}$ indicating which creases in $E$ bend convexly (mountain) or concavely (valley), we may \emph{flip} a face $F$ of $G$ to create a new MV assignment $\mu_F$ which equals $\mu$ except for all creases $e$ bordering $F$, where we have $\mu_F(e)=-\mu(e)$. In this paper we explore the configuration space of face flips for a variety of crease patterns $G$ that are tilings of the plane, proving examples where $\mu_F$ results in a MV assignment that is either never, sometimes, or always flat-foldable for various choices of $F$. We also consider the problem of finding, given two foldable MV assignments $\mu_1$ and $\mu_2$ of a given crease pattern $G$, a minimal sequence of face flips to turn $\mu_1$ into $\mu_2$. We find polynomial-time algorithms for this in the cases where $G$ is either a square grid or the Miura-ori, and show that this problem is NP-hard in the case where $G$ is the triangle lattice.

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., Comment: 15 pages

We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We show that this problem may be solved in linear time in trees, and is fixed-parameter tractable when parameterized either by the cyclomatic number of the input graph or by the length of the path. However, it is PSPACE-complete for paths of unbounded length in graphs of bounded bandwidth., Comment: 24 pages, 15 figures. Extended version of a paper to appear in the Proceedings of the 16th Algorithms and Data Structures Symposium (WADS 2019)

The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved., Comment: 37 pages, Discrete Mathematics and Theoretical Computer Science, DMTCS vol. 24:2, 2022, #9

$\Theta_6$-Graphs 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. Babu et al. (2014) conjectured that any $\Theta_6$-graph has a (near-)perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to $(3n-8)/7$. We also relate the size of maximum matchings in $\Theta_6$-graphs to the minimum size of a blocking set. Every edge of a $\Theta_6$-graph on point set $P$ corresponds to an empty triangle that contains the endpoints of the edge but no other point of $P$. A blocking set has at least one point in each such triangle. We prove that the size of a maximum matching is at least $\beta(n)/2$ where $\beta(n)$ is the minimum, over all theta-six graphs with $n$ vertices, of the minimum size of a blocking set. In the other direction, lower bounds on matchings can be used to prove bounds on $\beta$, allowing us to show that $\beta(n)\geq 3n/4-2$.

A \emph{Stick graph} is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a `ground line,' a line with slope $-1$. It is an open question to decide in polynomial time whether a given bipartite graph $G$ with bipartition $A\cup B$ has a Stick representation where the vertices in $A$ and $B$ correspond to horizontal and vertical segments, respectively. We prove that $G$ has a Stick representation if and only if there are orderings of $A$ and $B$ such that $G$'s bipartite adjacency matrix with rows $A$ and columns $B$ excludes three small `forbidden' submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of $A$ and $B$ permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of $A$ is given, we present an $O(|A|^3|B|^3)$-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable., Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)

We introduce a problem called the Minimum Shared-Power Edge Cut (MSPEC). The input to the problem is an undirected edge-weighted graph with distinguished vertices s and t, and the goal is to find an s-t cut by assigning "powers" at the vertices and removing an edge if the sum of the powers at its endpoints is at least its weight. The objective is to minimize the sum of the assigned powers. MSPEC is a graph generalization of a barrier coverage problem in a wireless sensor network: given a set of unit disks with centers in a rectangle, what is the minimum total amount by which we must shrink the disks to permit an intruder to cross the rectangle undetected, i.e. without entering any disc. This is a more sophisticated measure of barrier coverage than the minimum number of disks whose removal breaks the barrier. We develop a fully polynomial time approximation scheme (FPTAS) for MSPEC. We give polynomial time algorithms for the special cases where the edge weights are uniform, or the power values are restricted to a bounded set. Although MSPEC is related to network flow and matching problems, its computational complexity (in P or NP-hard) remains open.

We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms., Comment: Preliminary version in Proc. WG 2018

We prove that the following problem is complete for the existential theory of the reals: Given a planar graph and a polygonal region, with some vertices of the graph assigned to points on the boundary of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region. This strengthens an NP-hardness result by Patrignani on extending partial planar graph drawings. Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals. The complexity of the problem is open in the case of a simply connected region. We also show that, even for integer input coordinates, it is possible that drawing a graph in a polygonal region requires some vertices to be placed at irrational coordinates. By contrast, the coordinates are known to be bounded in the special case of a convex region, or for drawing a path in any polygonal region., Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence, that is increasing or decreasing, has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as a polygonal path for which every maximal sub-path, with positive- or negative-slope edges, has at least three points. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximum-length subsequence that is a rollercoaster. It was conjectured that every sequence of $n$ distinct real numbers contains a rollercoaster of length at least $\lceil n/2\rceil$ for $n>7$, while the best known lower bound is $\Omega(n/\log n)$. In this paper we prove this conjecture. Our proof is constructive and implies a linear-time algorithm for computing a rollercoaster of this length. Extending the $O(n\log n)$-time algorithm for computing a longest increasing subsequence, we show how to compute a maximum-length rollercoaster within the same time bound. A maximum-length rollercoaster in a permutation of $\{1,\dots,n\}$ can be computed in $O(n \log \log n)$ time. The search for rollercoasters was motivated by orthogeodesic point-set embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A top-view caterpillar is one of degree 4 such that the two leaves adjacent to each vertex lie on opposite sides of the spine. As an application of our result on rollercoasters, we are able to find a planar drawing of every $n$-node top-view caterpillar on every set of $\frac{25}{3}n$ points in the plane, such that each edge is an orthogonal path with one bend. This improves the previous best known upper bound on the number of required points, which is $O(n \log n)$. We also show that such a drawing can be obtained in linear time, provided that the points are given in sorted order., Comment: 17 pages

A geometric graph in the plane is angle-monotone of width $\gamma$ if every pair of vertices is connected by an angle-monotone path of width $\gamma$, a path such that the angles of any two edges in the path differ by at most $\gamma$. Angle-monotone graphs have good spanning properties. We prove that every point set in the plane admits an angle-monotone graph of width $90^\circ$, hence with spanning ratio $\sqrt 2$, and a subquadratic number of edges. This answers an open question posed by Dehkordi, Frati and Gudmundsson. We show how to construct, for any point set of size $n$ and any angle $\alpha$, $0 < \alpha < 45^\circ$, an angle-monotone graph of width $(90^\circ+\alpha)$ with $O(\frac{n}{\alpha})$ edges. Furthermore, we give a local routing algorithm to find angle-monotone paths of width $(90^\circ+\alpha)$ in these graphs. The routing ratio, which is the ratio of path length to Euclidean distance, is at most $1/\cos(45^\circ + \frac{\alpha}{2})$, i.e., ranging from $\sqrt 2 \approx 1.414$ to $2.613$. For the special case $\alpha = 30^\circ$, we obtain the $\Theta_6$-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width $120^\circ$.

Given a triangulation of a point set in the plane, a \emph{flip} deletes an edge $e$ whose removal leaves a convex quadrilateral, and replaces $e$ by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips. We characterize when this is possible by proving the \emph{Orbit Conjecture} of Bose, Lubiw, Pathak and Verdonschot which states that \emph{all} labels can be simultaneously mapped to their destination if and only if \emph{each} label individually can be mapped to its destination. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of $O(n^7)$ on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the \emph{flip complex}, has the usual flip graph as its $1$-skeleton. We use properties of the $2$-skeleton of the flip complex to prove the Orbit Conjecture., Comment: 19 pages, 2 figures

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. Drawing ordered trees with L-shaped edges is harder---we give an example that cannot be done and a bound of $O(n \log n)$ points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars., Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017)

We reprove a result of Dehkordi, Frati, and Gudmundsson: every two vertices in a non-obtuse triangulation of a point set are connected by an angle-monotone path--an xy-monotone path in an appropriately rotated coordinate system. We show that this result cannot be extended to angle-monotone spanning trees, but can be extended to boundary-rooted spanning forests. The latter leads to a conjectural edge-unfolding of sufficiently shallow polyhedral convex caps., Comment: 6 pages, 9 figures, 6 references. To appear in the *Canadian Conference on Computational Geometry*, July 2017

Two vertex-labelled polygons are \emph{compatible} if they have the same clockwise cyclic ordering of vertices. The definition extends to polygonal regions (polygons with holes) and to triangulations---for every face, the clockwise cyclic order of vertices on the boundary must be the same. It is known that every pair of compatible $n$-vertex polygonal regions can be extended to compatible triangulations by adding $O(n^2)$ Steiner points. Furthermore, $\Omega(n^2)$ Steiner points are sometimes necessary, even for a pair of polygons. Compatible triangulations provide piecewise linear homeomorphisms and are also a crucial first step in morphing planar graph drawings, aka "2D shape animation". An intriguing open question, first posed by Aronov, Seidel, and Souvaine in 1993, is to decide if two compatible polygons have compatible triangulations with at most $k$ Steiner points. In this paper we prove the problem to be NP-hard for polygons with holes. The question remains open for simple polygons., Comment: A preliminary version appeared at the 29th Canadian Conference on Computational Geometry (CCCG 2017)

We consider the conjecture by Aichholzer, Aurenhammer, Hurtado, and Krasser that any two points sets with the same cardinality and the same size convex hull can be triangulated in the "same" way, more precisely via \emph{compatible triangulations}. We show counterexamples to various strengthened versions of this conjecture.

For a matroid with an ordered (or "labelled") basis, a basis exchange step removes one element with label $l$ and replaces it by a new element that results in a new basis, and with the new element assigned label $l$. We prove that one labelled basis can be reconfigured to another if and only if for every label, the initial and final elements with that label lie in the same connected component of the matroid. Furthermore, we prove that when the reconfiguration is possible, the number of basis exchange steps required is $O(r^{1.5})$ for a rank $r$ matroid. For a graphic matroid we improve the bound to $O(r \log r)$.

We present two universal hinge patterns that enable a strip of material to fold into any connected surface made up of unit squares on the 3D cube grid--for example, the surface of any polycube. The folding is efficient: for target surfaces topologically equivalent to a sphere, the strip needs to have only twice the target surface area, and the folding stacks at most two layers of material anywhere. These geometric results offer a new way to build programmable matter that is substantially more efficient than what is possible with a square $N \times N$ sheet of material, which can fold into all polycubes only of surface area $O(N)$ and may stack $\Theta(N^2)$ layers at one point. We also show how our strip foldings can be executed by a rigid motion without collisions (albeit assuming zero thickness), which is not possible in general with 2D sheet folding. To achieve these results, we develop new approximation algorithms for milling the surface of a grid polyhedron, which simultaneously give a 2-approximation in tour length and an 8/3-approximation in the number of turns. Both length and turns consume area when folding a strip, so we build on past approximation algorithms for these two objectives from 2D milling., Comment: 22 pages, 18 figures. Added figures and clarified text

A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is $x$- and $y$-monotone. Angle-monotone graphs are $\sqrt 2$-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-$\theta_6$-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex $s$ to any vertex $t$ whose length is within $1 + \sqrt 2$ times the Euclidean distance from $s$ to $t$. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations., Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016)

Given an $n$-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of $O(n)$ steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns' 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps., Comment: 31 pages, 18 figures

Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many algorithms and properties of shortest paths and convex hulls in Euclidean space fail to transfer over. We give an algorithm, using linear programming, to compute the convex hull of a set of points in a 2-dimensional CAT(0) polyhedral complex with a single vertex. We explore the use of shortest path maps to answer single-source shortest path queries in 2-dimensional CAT(0) polyhedral complexes, and we unify efficient solutions for 2-manifold and rectangular cases., Comment: minor clarifications

We study versions of cop and robber pursuit-evasion games on the visibility graphs of polygons, and inside polygons with straight and curved sides. Each player has full information about the other player's location, players take turns, and the robber is captured when the cop arrives at the same point as the robber. In visibility graphs we show the cop can always win because visibility graphs are dismantlable, which is interesting as one of the few results relating visibility graphs to other known graph classes. We extend this to show that the cop wins games in which players move along straight line segments inside any polygon and, more generally, inside any simply connected planar region with a reasonable boundary. Essentially, our problem is a type of pursuit-evasion using the link metric rather than the Euclidean metric, and our result provides an interesting class of infinite cop-win graphs., Comment: 23 pages

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation substitutes a vertex $v$ by at most $k$ new vertices such that each neighbor of $v$ is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are $2$-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify $k$-splittability in linear time, for a constant $k$.

A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix $M$ that has a $1$ in the entry $(i,j)$ iff there is a path from the $j$th source to the $i$th sink. The smallest number of edges in a rectifier network computing $M$ is a classic complexity measure on matrices, which has been studied for more than half a century. We explore two well-known techniques that have hitherto found little to no applications in this theory. Both of them build on a basic fact that depth-$2$ rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. We obtain new results by using fractional and greedy coverings (defined in the standard way). First, we show that all fractional coverings of the so-called full triangular matrix have cost at least $n\log n$. This provides (a fortiori) a new proof of the tight lower bound on its depth-$2$ complexity (the exact value has been known since 1965, but previous proofs are based on different arguments). Second, we show that the greedy heuristic is instrumental in tightening the upper bound on the depth-$2$ complexity of the Kneser-Sierpi\'nski (disjointness) matrix. The previous upper bound is $O(n^{1.28})$, and we improve it to $O(n^{1.17})$, while the best known lower bound is $\Omega(n^{1.16})$. Third, using fractional coverings, we obtain a form of direct product theorem that gives a lower bound on unbounded-depth complexity of Kronecker (tensor) products of matrices. In this case, the greedy heuristic shows (by an argument due to Lov\'asz) that our result is only a logarithmic factor away from the "full" direct product theorem. Our second and third results constitute progress on open problem 7.3 and resolve, up to a logarithmic factor, open problem 7.5 from a recent book by Jukna and Sergeev (in Foundations and Trends in Theoretical Computer Science (2013)).

We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case., Comment: To appear in SoCG 2015

In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative fold orientations), but in general this specification is consistent with exponentially many possible folded states. We analyze the complexity of finding the best consistent folded state according to two metrics: minimizing the total number of layers in the folded state (so that a "flat folding" is indeed close to flat), and minimizing the total amount of paper required to execute the folding (where "thicker" creases consume more paper). We prove both problems strongly NP-complete even for 1D folding. On the other hand, we prove the first problem fixed-parameter tractable in 1D with respect to the number of layers., Comment: 9 pages, 7 figures

Alamdari et al. showed that given two straight-line planar drawings of a graph, there is a morph between them that preserves planarity and consists of a polynomial number of steps where each step is a \emph{linear morph} that moves each vertex at constant speed along a straight line. An important step in their proof consists of converting a \emph{pseudo-morph} (in which contractions are allowed) to a true morph. Here we introduce the notion of \emph{unidirectional morphing} step, where the vertices move along lines that all have the same direction. Our main result is to show that any planarity preserving pseudo-morph consisting of unidirectional steps and contraction of low degree vertices can be turned into a true morph without increasing the number of steps. Using this, we strengthen Alamdari et al.'s result to use only unidirectional morphs, and in the process we simplify the proof., Comment: 13 pages, 9 figures

We consider the problem of morphing between two planar drawings of the same triangulated graph, maintaining straight-line planarity. A paper in SODA 2013 gave a morph that consists of $O(n^2)$ 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., Comment: 23 pages, 8 figures

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, a constant number of crossings between every pair of edges. 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., Comment: Preliminary version appeared at the 22nd International Symposium on Graph Drawing (GD '14)

When can a plane graph with prescribed edge lengths and prescribed angles (from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states., Comment: 21 pages, 10 figures

Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue's universal cover problem and force closure grasps. In addition to considering special shapes (squares, equilateral triangles, polygons and disks), we give upper and lower bounds on scale factors for single folds of convex objects and arbitrary folds of simply connected objects., Comment: 19 pages, 10 figures, to appear in Journal of Computational Geometry (JoCG)

Consider an orthogonal polyhedron, i.e., a polyhedron where (at least after a suitable rotation) all faces are perpendicular to a coordinate axis, and hence all edges are parallel to a coordinate axis. Clearly, any facial angle and any dihedral angle is a multiple of $\pi/2$. In this note we explore the converse: if the facial and/or dihedral angles are all multiples of $\pi /2$, is the polyhedron necessarily orthogonal? The case of facial angles was answered previously. In this note we show that if both the facial and dihedral angles are multiples of $\pi /2$ then the polyhedron is orthogonal (presuming connectivity), and we give examples to show that the condition for dihedral angles alone does not suffice., Comment: 3 pages

We study a geometric representation problem, where we are given a set $\cal R$ of axis-aligned rectangles with fixed dimensions and a graph with vertex set $\cal R$. The task is to place the rectangles without overlap such that two rectangles touch if and only if the graph contains an edge between them. We call this problem Contact Representation of Word Networks (CROWN). It formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Here, we represent words by rectangles and semantic relationships by edges. We show that CROWN is strongly NP-hard even restricted trees and weakly NP-hard if restricted stars. We consider the optimization problem Max-CROWN where each adjacency induces a certain profit and the task is to maximize the sum of the profits. For this problem, we present constant-factor approximations for several graph classes, namely stars, trees, planar graphs, and graphs of bounded degree. Finally, we evaluate the algorithms experimentally and show that our best method improves upon the best existing heuristic by 45%., Comment: 14 pages, 8 figures. arXiv admin note: text overlap with arXiv:1304.8016

Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge $e$, which is the set of all edges reachable from $e$ via flips. We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight $\Theta(n \log n)$ bounds for edge-labelled triangulations of $n$-vertex convex polygons and combinatorial triangulations, contrasting with the $\Theta(n)$ bounds in their respective unlabelled settings. The $\Omega(n \log n)$ lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to $O(\log^2 n)$, although we no longer have a matching lower bound. Moving beyond convex polygons, we show that edge-labelled triangulated polygons with a single reflex vertex can have a disconnected flip graph. This is in sharp contrast with the unlabelled case, where the flip graph is connected for any triangulated polygon. For spiral polygons, we provide a complete characterization of the orbits. This allows us to decide connectivity of the flip graph of a spiral polygon in linear time. We also prove an upper bound of $O(n^2)$ on the diameter of each connected component, which is optimal in the worst case. We conclude with an example of a non-spiral polygon whose flip graph has diameter $\Omega(n^3)$., Comment: Significantly expanded journal version

In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner. We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3) constructing a self-approaching Steiner network connecting a given set of points. We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in R^2 and R^3, but it is NP-hard to test if a given graph drawing in R^3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.

Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the plane.

The Rubik's Cube is perhaps the world's most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik's Cube also has a rich underlying algorithmic structure. Specifically, we show that the n x n x n Rubik's Cube, as well as the n x n x 1 variant, has a "God's Number" (diameter of the configuration space) of Theta(n^2/log n). The upper bound comes from effectively parallelizing standard Theta(n^2) solution algorithms, while the lower bound follows from a counting argument. The upper bound gives an asymptotically optimal algorithm for solving a general Rubik's Cube in the worst case. Given a specific starting state, we show how to find the shortest solution in an n x O(1) x O(1) Rubik's Cube. Finally, we show that finding this optimal solution becomes NP-hard in an n x n x 1 Rubik's Cube when the positions and colors of some of the cubies are ignored (not used in determining whether the cube is solved)., Comment: 34 pages, 9 figures. A short version of this paper is to appear at the 19th Annual European Symposium on Algorithms

Two planar graphs G1 and G2 sharing some vertices and edges are `simultaneously planar' if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them., Comment: Appeared in ISAAC 2010, 15 pages, 3 figures