Let G be a directed graph with n vertices and non-negative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe a randomized algorithm to preprocess the graph in O(g n log n) time with high probability, so that the shortest-path distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)-time algorithm of Klein [Multiple-source shortest paths in planar graphs. In Proc. 16th Ann. ACM-SIAM Symp. Discrete Algorithms, 2005] for multiple-source shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortest-path tree as its source point moves continuously around the boundary of f . As an application of our algorithm, we describe algorithms to compute a shortest non-contractible or non-separating cycle in embedded, undirected graphs in O(g 2 n log n) time with high probability. Our high-probability time bounds hold in the worst-case for generic edge weights, or with an additional O(log n) factor for arbitrary edge weights.

We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2 ) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε ), this represents a considerable improvement over previous results by Thomassen, and Erickson and HarPeled. We also give algorithms to find a shortest non-contractible cycle in O(g O(g) V 3/2 ) time, which improves previous results for fixed genus.This result can be applied for computing the (non-separating) face-width of embedded graphs. Using similar ideas we provide the first near-linear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the face-width of embedded toroidal graphs in O(V 5/4 log V ) time.

A graph is near-planar if it can be obtained from a planar graph by adding an edge. We show the surprising fact that it is NP-hard to compute the crossing number of near-planar graphs. A graph is 1-planar if it has a drawing where every edge is crossed by at most one other edge. We show that it is NP-hard to decide whether a given near-planar graph is 1-planar. The main idea in both reductions is to consider the problem of simultaneously drawing two planar graphs inside a disk, with some of its vertices fixed at the boundary of the disk. This leads to the concept of anchored embedding, which is of independent interest. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný. Introduction.A drawing of a graph G in the plane is a representation of G where vertices are represented by distinct points of R 2 , edges are represented by simple polygonal arcs in R 2 joining points that correspond to their endvertices, and the interior of every arc representing an edge contains no points representing the vertices of G. A crossing of a drawing D is a pair ({e, e }, p), where e and e are distinct edges and p ∈ R 2 is a point that belongs to the interiors of both arcs representing e and e in the drawing D. The number of crossings of a drawing D is denoted by cr(D) and is called the crossing number of the drawing. The crossing number cr(G) of a graph G is the minimum cr(D) taken over all drawings D of G. A planar graph is a graph whose crossing number is 0. A drawing D with cr(D) = 0 is called an embedding of G (in the plane). A drawing D is a 1-drawing if each edge participates in at most one crossing. A 1-planar graph is a graph that has some 1-drawing.A graph is near-planar if it can be obtained from a planar graph G by adding an extra edge xy between vertices x and y of G. We denote such a near-planar graph by G + xy. (The term almost planar has also been used for the same concept [17,21].) Near-planarity is a very weak relaxation of planarity, and hence it is natural to study properties of near-planar graphs. Graphs embeddable in the torus and apex graphs are superfamilies of near-planar graphs.We show that it is NP-hard to compute the crossing number of near-planar graphs. We also show that it is NP-hard to decide whether a given near-planar graph is

We show how to compute in O(n 11/6 polylog(n)) expected time the diameter and the sum of the pairwise distances in an undirected planar graph with n vertices and positive edge weights. These are the first algorithms for these problems using time O(n c ) for some constant c < 2.

Let K be a simplicial complex and g the rank of its p-th homology group H p (K) defined with Z 2 coefficients. We show that we can compute a basis H of H p (K) and annotate each p-simplex of K with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω ) time, where n is the size of K and ω < 2.376 is a quantity so that two n × n matrices can be multiplied in O(n ω ) time. The pre-computation of annotations permits answering queries about the independence or the triviality of p-cycles efficiently.

Let G be a unit disk graph in the plane defined by n disks whose positions are known. For the case when G is unweighted, we give a simple algorithm to compute a shortest path tree from a given source in O(n log n) time. For the case when G is weighted, we show that a shortest path tree from a given source can be computed in O(n 1+ε ) time, improving the previous best time bound of O(n 4/3+ε ).

No abstract

We show how to compute in O(n 4/3 log 1/3 n + n 2/3 k 2/3 log n) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/ log n) 5/6 ≤ k ≤ n 2 / log 6 n. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs.