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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 $n-3$.

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

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., Preliminary version in Proc. WG 2018

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), constant-factor approximation algorithms, and some poly-time exact algorithms for simple graph classes such as cliques, stars, paths, and cycles. Sequential and parallel token swapping on 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., LIPIcs, Vol. 244, 30th Annual European Symposium on Algorithms (ESA 2022), pages 3:1-3:15

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

$��_6$-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 $��_6$-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 $��_6$-graphs to the minimum size of a blocking set. Every edge of a $��_6$-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 $��(n)/2$ where $��(n)$ 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 $��$, allowing us to show that $��(n)\geq 3n/4-2$.

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.

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 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

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)

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 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.

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.

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

In a recent paper, we introduced the simultaneous representation problem (defined for any graph class C) and studied the problem for chordal, comparability and permutation graphs. For interval graphs, the problem is defined as follows. Two interval graphs G_1 and G_2, sharing some vertices I (and the corresponding induced edges), are said to be `simultaneous interval graphs' if there exist interval representations R_1 and R_2 of G_1 and G_2, such that any vertex of I is mapped to the same interval in both R_1 and R_2. Equivalently, G_1 and G_2 are simultaneous interval graphs if there exist edges E' between G_1-I and G_2-I such that G_1 \cup G_2 \cup E' is an interval graph. Simultaneous representation problems are related to simultaneous planar embeddings, and have applications in any situation where it is desirable to consistently represent two related graphs, for example: interval graphs capturing overlaps of DNA fragments of two similar organisms; or graphs connected in time, where one is an updated version of the other. In this paper we give an O(n^2*logn) time algorithm for recognizing simultaneous interval graphs,where n = |G_1 \cup G_2|. This result complements the polynomial time algorithms for recognizing probe interval graphs and provides an efficient algorithm for the interval graph sandwich problem for the special case where the set of optional edges induce a complete bipartite graph.

In this paper we introduce the 'simultaneous membership problem', defined for any graph class C characterized in terms of representations, e.g. any class of intersection graphs. Two graphs G_1 and G_2, sharing some vertices X (and the corresponding induced edges), are said to be 'simultaneous members' of graph class C, if there exist representations R_1 and R_2 of G_1 and G_2 that are "consistent" on X. Equivalently (for the classes C that we consider) there exist edges E' between G_1-X and G_2-X such that G_1 \cup G_2 \cup E' belongs to class C. Simultaneous membership problems have application in any situation where it is desirable to consistently represent two related graphs, for example: interval graphs capturing overlaps of DNA fragments of two similar organisms; or graphs connected in time, where one is an updated version of the other. Simultaneous membership problems are related to simultaneous planar embeddings, graph sandwich problems and probe graph recognition problems. In this paper we give efficient algorithms for the simultaneous membership problem on chordal, comparability and permutation graphs. These results imply that graph sandwich problems for the above classes are tractable for an interesting special case: when the set of optional edges form a complete bipartite graph. Our results complement the recent polynomial time recognition algorithms for probe chordal, comparability, and permutation graphs, where the set of optional edges form a clique., Comment: 15 pages, 1 figure

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We give a simple approximation algorithm that solves the SDP problem on general terrains. Our algorithm discretizes the terrain with O(n^2 X / e) Steiner points so that after an O(n^2 X / e * log(n X /e))-time preprocessing phase for a given vertex s, we can determine a (1+e)-approximate SDP from s to any point v in O(n) time if v is either a vertex of the terrain or a Steiner point, and in O(n X /e) time otherwise. Here n is the size of the terrain, and X is a parameter of the geometry of the terrain., Comment: 14 pages, 3 figures

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is polynomial. There are, however, simple polygons with an exponential number of distinct gluings. In the reverse direction, we show that there are polytopes with an exponential number of distinct cuttings that lead to simple unfoldings. We establish necessary conditions for a polytope to have convex unfoldings, implying, for example, that among the Platonic solids, only the tetrahedron has a convex unfolding. We provide an inventory of the polytopes that may unfold to regular polygons, showing that, for n>6, there is essentially only one class of such polytopes., 54 pages, 33 figures

A bipartite graph \(G=(A,B,E)\) is \(\mathcal{H}\)-convex, for some family of graphs \(\mathcal{H}\), if there exists a graph \(H\in \mathcal{H}\) with \(V(H)=A\) such that the set of neighbours in A of each \(b\in B\) induces a connected subgraph of H. Many \(\mathsf {NP}\)-complete problems become polynomial-time solvable for \({\mathcal H}\)-convex graphs when \({\mathcal H}\) is the set of paths. In this case, the class of \({\mathcal H}\)-convex graphs is known as the class of convex graphs. The underlying reason is that this class has bounded mim-width. We extend the latter result to families of \({\mathcal H}\)-convex graphs where (i) \({\mathcal H}\) is the set of cycles, or (ii) \({\mathcal H}\) is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we can re-prove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of \({\mathcal H}\)-convex graphs is unbounded if \({\mathcal H}\) is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of \(\mathcal{H}\)-convex graphs.

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in R^3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(n^(1/2)). A matching Omega(n^(1/2)) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log(n)) upper bound of Aronov et al. [Aronov et al., in proc. of LATIN, 2006] for farthest-point Voronoi diagrams in the special case where points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of S U (p) from the diagram of S, in time O(K polylog n) worst case, which is O(n^(1/2) polylog n) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation., Leibniz International Proceedings in Informatics (LIPIcs), 51, ISSN:1868-8969, 32nd International Symposium on Computational Geometry (SoCG 2016), ISBN:978-3-95977-009-5

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)

For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a minimum $st$-cut in $G$? We solve this problem in preprocessing time $O(n\log^3 n)$ for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space $O(n)$ that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is $2^{O(g^2)}$.