Abstract. We consider the embeddability problem of a graph [Formula: see text] into a two-dimensional simplicial complex [Formula: see text]: Given [Formula: see text] and [Formula: see text], decide whether [Formula: see text] admits a topological embedding into [Formula: see text]. The problem is NP-hard, even in the restricted case where [Formula: see text] is homeomorphic to a surface. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing an [Formula: see text]-time algorithm. If [Formula: see text] embeds into [Formula: see text], we can compute a representation of an embedding in the same amount of time. Moreover, we show that several known problems reduce to this one, such as the crossing number and the planarity number problems, and, under some conditions, the embedding extension problem. Our approach is to reduce to the case where [Formula: see text] has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface, but only on algorithms from graph minor theory. However, by combining our results with a linear-time algorithm for embedding graphs on surfaces and with a very recent result for the irrelevant vertex method, we can decide whether [Formula: see text] embeds into [Formula: see text] in [Formula: see text] time, for some function [Formula: see text].

Tutte's celebrated barycentric embedding theorem describes a natural way to build straight-line embeddings (crossing-free drawings) of a (3-connected) planar graph: map the vertices of the outer face to the vertices of a convex polygon, and ensure that each remaining vertex is in convex position, namely, a barycenter with positive coefficients of its neighbors. Actually computing an embedding then boils down to solving a system of linear equations. A particularly appealing feature of this method is the flexibility given by the choice of the barycentric weights. Generalizations of Tutte's theorem to surfaces of nonpositive curvature are known, but due to their inherently continuous nature, they do not lead to an algorithm. In this paper, we propose a purely discrete analog of Tutte's theorem for surfaces (with or without boundary) of nonpositive curvature, based on the recently introduced notion of reducing triangulations. We prove a Tutte theorem in this setting: every drawing homotopic to an embedding such that each vertex is harmonious (a discrete analog of being in convex position) is a weak embedding (arbitrarily close to an embedding). We also provide a polynomial-time algorithm to make an input drawing harmonious without increasing the length of any edge, in a similar way as a drawing can be put in convex position without increasing the edge lengths. 48 pages. This is the TheoretiCS journal version

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Consider a graph drawn on a surface (for example, the plane minus a finite set of obstacle points), possibly with crossings. We provide an algorithm to decide whether such a drawing can be untangled, namely, if one can slide the vertices and edges of the graph on the surface (avoiding the obstacles) to remove all crossings; in other words, whether the drawing is homotopic to an embedding. While the problem boils down to planarity testing when the surface is the sphere or the disk (or equivalently the plane without any obstacle), the other cases have never been studied before, except when the input graph is a cycle, in an abundant literature in topology and more recently by Despré and Lazarus [SoCG 2017, J. ACM 2019], who gave a near-linear algorithm for this problem.

Geometry (SoCG) was held online, June 7-11, 2021, as part of the Computational Geometry Week.This special issue of Discrete & Computational Geometry contains a selection of papers from the symposium.Of the 164 submissions to SoCG'21, 58 were accepted by the Program Committee.Of these, a selection of six especially strong papers are included in this issue.They were submitted, refereed, and revised according to the usual high standards of D&CG.We thank the authors of all submitted papers for revising and polishing their work.We are grateful to the anonymous referees for their dedication and expertise that ensure the high quality of the articles in this special issue.The articles appear in this special issue in alphabetical order of the names of the first authors.In the remainder of this foreword we briefly introduce each paper in the same order.Interval graphs form one of the most fundamental classes of geometric intersection graphs.They are well structured, which makes many problems that are NP-hard for general graphs polynomial-time solvable on intersection graphs.Ranendu Adhikary, Kaustav Bose, Satwik Mukherjee, and Bodhayan Roy show that this is not the case for the maximum cut problem.They prove that this problem is NP-complete for interval graphs, solving a long-standing open problem that was already posed more than 35 years ago in Johnson's NP-completeness column.The next paper returns to a classic problem in computational geometry: given a set of n points in d-dimensional space and given a bounding box, compute the largest (volume-wise) box inside the bounding box that does not contain any of the input points.Timothy Chan presents improved algorithms for this problem for d = 2, d = 3, and d ≥ 4. For d = 2, a clever combination of interval trees, lower envelopes, and pseudo-lines are used to reduce the problem to subproblems of logarithmic size.This results in a time bound of O(n2 O(log * n) log n).The previous best bound was

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus G has a cut graph of length at most a given value. We prove a time lower bound for this problem of n Ω( g log g ) conditionally to the ETH. In other words, the first n O(g) -time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year-old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals , is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n Ω( gt + g 2 + t log ( g + t )) , conditionally to the ETH, for any choice of the genus g ≥ 0 of the graph and the number of terminals t ≥ 4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a gridlike structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value G of the genus.

For an undirected edge-weighted graph $G$ and a set $R$ of pairs of vertices called pairs of terminals, a multicut is a set of edges such that removing these edges from $G$ disconnects each pair in $R$. We provide an algorithm computing a $(1+\varepsilon)$-approximation of the minimum multicut of a graph $G$ in time $(g+t)^{(O(g+t)^3)}\cdot(1/\varepsilon)^{O(g+t)} \cdot n \log n$, where $g$ is the genus of $G$ and $t$ is the number of terminals. This is tight in several aspects, as the minimum multicut problem is both APX-hard and W[1]-hard (parameterized by the number of terminals), even on planar graphs (equivalently, when $g=0$). Our result, in the field of fixed-parameter approximation algorithms, mostly relies on concepts borrowed from computational topology of graphs on surfaces. In particular, we use and extend various recent techniques concerning homotopy, homology, and covering spaces. Interestingly, such topological techniques seem necessary even for the planar case. We also exploit classical ideas stemming from approximation schemes for planar graphs and low-dimensional geometric inputs. A key insight toward our result is a novel characterization of a minimum multicut as the union of some Steiner trees in the universal cover of the surface in which $G$ is embedded.

We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph G embedded on a surface S is a subgraph of G whose removal from S leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus g has a cut graph of length at most a given value. We prove a time lower bound for this problem of n^{Omega(g/log g)} conditionally to ETH. In other words, the first n^{O(g)}-time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year old question of these authors. A multiway cut of an undirected graph G with t distinguished vertices, called terminals, is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph G has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of n^{Omega(sqrt{gt + g^2}/log(gt))}, conditionally to ETH, for any choice of the genus g >=0 of the graph and the number of terminals t >=4. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a grid-like structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value g of the genus.

We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the topological crossing number of a graph, but is more general. The problem is NP-complete when C is part of the input, and we give a polynomial-time algorithm if the complex C is fixed. Our strategy is to reduce the problem to an embedding extension problem on a surface, which has the following form: Given a subgraph H' of a graph G', and an embedding of H' on a surface S, can that embedding be extended to an embedding of G' on S? Such problems can be solved, in turn, using a key component in Mohar's algorithm to decide the embeddability of a graph on a fixed surface (STOC 1996, SIAM J. Discr. Math. 1999).

For an undirected edge-weighted graph G and a set R of pairs of vertices called pairs of terminals, a multicut is a set of edges such that removing these edges from G disconnects each pair in R. We provide an algorithm computing a (1 + ε)-approximation of the minimum multicut of a graph G in time (g + t)(O(g+t)3) · (1/ε)°(g+t) · n log n, where g is the genus of G and t is the number of terminals. This is tight in several aspects, as the minimum multicut problem is both APX-hard and W[1]-hard (parameterized by the number of terminals), even on planar graphs (equivalently, when g = 0). Our result, in the field of fixed-parameter approximation algorithms, mostly relies on concepts borrowed from computational topology of graphs on surfaces. In particular, we use and extend various recent techniques concerning homotopy, homology, and covering spaces (even in the planar case). We also exploit classical ideas stemming from approximation schemes for planar graphs and low-dimensional geometric inputs. A key insight towards our result is a novel characterization of a minimum multicut as the union of some Steiner trees in the universal cover of the surface in which G is embedded.

We present an algorithm for the following problem. Given a triangulated 3-manifold M and a (possibly non-simple) closed curve on the boundary of M, decide whether this curve is contractible in M. Our algorithm is combinatorial and runs in exponential time. This is the first algorithm that is specifically designed for this problem; its running time considerably improves upon the existing bounds implicit in the literature for the more general problem of contractibility of closed curves in a 3-manifold. The proof of the correctness of the algorithm relies on methods of 3-manifold topology and in particular on those used in the proof of the Loop Theorem.

We present a framework for addressing several problems on weighted planar graphs and graphs of bounded genus. With that framework, we derive polynomial-time approximation schemes for the following problems in planar graphs or graphs of bounded genus: edge-weighted tree cover and tour cover; vertex-weighted connected dominating set, max-weight-leaf spanning tree, and connected vertex cover. In addition, we obtain a polynomial-time approximation scheme for feedback vertex set in planar graphs. These are the first polynomial-time approximation schemes for all those problems in weighted embedded graphs. (For unweighted versions of some of these problems, polynomial-time approximation schemes were previously given using bidimensionality.)

How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).

We investigate the following problem: Given two embeddings G1 and G2 of the same abstract graph G on an orientable surface S, decide whether G1 and G2 are isotopic; in other words, whether there exists a continuous family of embeddings between G1 and G2. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G1 (resp., G2) with a fixed graph cellularly embedded on S; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G1 and G2 are piecewise-linear embeddings in the plane minus a finite set of points; our algorithm runs in O(n3/2log n) time, where n is the complexity of the input.

The nerve of a family of sets is a simplicial complex that records the intersection pattern of its subfamilies. Nerves are widely used in computational geometry and topology, because the nerve theorem guarantees that the nerve of a family of geometric objects has the same topology as the union of the objects, if they form a good cover. In this paper, we relax the good cover assumption to the case where each subfamily intersects in a disjoint union of possibly several homology cells, and we prove a generalization of the nerve theorem in this framework, using spectral sequences from algebraic topology. We then deduce a new topological Helly-type theorem that unifies previous results of Amenta, Kalai and Meshulam, and Matousek. This Helly-type theorem is used to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory.

Let G be a graph cellularly embedded on a surface $\mathcal{S}$. We consider the problem of determining whether G contains a cycle (i.e., a closed walk without repeated vertices) of a certain topological type in $\mathcal{S}$. We show that the problem can be answered in linear time when the topological type is one of the following: contractible, noncontractible, or nonseparating. In each case, we obtain the same time complexity if we require the cycle to contain a given vertex. On the other hand, we prove that the problem is NP-complete when considering separating or splitting cycles. We also show that deciding the existence of a separating or a splitting cycle of length at most k is fixed-parameter tractable with respect to k plus the genus of the surface.

Let G be a directed planar graph of complexity n , each arc having a nonnegative length. Let s and t be two distinct faces of G let s 1 ,…, s k be vertices incident with s let t 1 ,…, t k be vertices incident with t . We give an algorithm to compute k pairwise vertex-disjoint paths connecting the pairs ( s i , t i ) in G , with minimal total length, in O ( kn log n ) time.

Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest non-contractible and a shortest non-separating cycle of G.

Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest non-contractible and a shortest surface non-separating cycle in D. This generalizes previous results that only dealt with undirected graphs.

We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity n, genus $g\geq2$, and no boundary, we construct in $O(gn\log n)$ time a tight octagonal decomposition of the surface—a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface into a complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity k in $O(gnk)$ time, or the shortest cycle homotopic to a given cycle of complexity k in $O(gnk\log(nk))$ time. A similar algorithm computes shortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colin de Verdière and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.

The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles ("trees"). We describe a polynomial-time algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of obstacles, which are either points or polygons.

Let $G$ be a directed planar graph of complexity~$n$, each arc having a nonnegative length. Let $s$ and~$t$ be two distinct faces of~$G$; let $s_1,ldots,s_k$ be vertices incident with~$s$; let $t_1,ldots,t_k$ be vertices incident with~$t$. We give an algorithm to compute $k$ pairwise vertex-disjoint paths connecting the pairs $(s_i,t_i)$ in~$G$, with minimal total length, in $O(knlog n)$ time.

We consider the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface. For this purpose, we use a pants decomposition of the surface: a set of disjoint simple cycles that cut the surface into pairs of pants (spheres with three holes). We solve this problem in a framework where the cycles are closed walks on the vertex-edge graph of a combinatorial surface that may overlap but do not cross. We give an algorithm that transforms an input pants decomposition into another homotopic pants decomposition that is optimal : each cycle is as short as possible in its homotopy class. As a consequence, finding a shortest cycle homotopic to a given simple cycle amounts to extending the cycle into a pants decomposition and to optimizing it: the resulting pants decomposition contains the desired cycle. We describe two algorithms for extending a cycle to a pants decomposition. All algorithms in this article are polynomial, assuming uniformity of the weights of the vertex-edge graph of the surface.

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This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi tessellation in a conformal parameter space, where the specified density function is used for weighing. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.

We describe an algorithm which, for any piecewise linear complex (PLC) in 3D, builds a Delaunay triangulation conforming to this PLC.The algorithm has been implemented, and yields in practice a relatively small number of Steiner points due to the fact that it adapts to the local geometry of the PLC. It is, to our knowledge, the first practical algorithm devoted to this problem.

Every compact orientable boundaryless surface /spl Mscr/ can be cut along simple loops with a common point /spl upsi//sub 0/, pairwise disjoint except at /spl upsi//sub 0/, so that the resulting surface is a topological disk; such a set of loops is called a fundamental system of loops for /spl Mscr/. The resulting disk is a polygon in which the edges are pairwise identified on the surface; it is called a polygonal schema Assuming that /spl Mscr/ is triangulated, and that each edge has a given length, we are interested in a shortest (or optimal) system homotopic to a given one, drawn on the vertex-edge graph of /spl Mscr/. We prove that each loop of such an optimal system is a shortest loop among all simple loops in its homotopy class. We give a polynomial (under some reasonable assumptions) algorithm to build such a system. As a byproduct, we get a polynomial algorithm to compute a shortest simple loop homotopic to a given simple loop.