Algorithms for Graphs Embeddable with Few Crossings per Edge

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We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Łącki and Sankowski, for any constant g. Indeed, our algorithm calls Łącki and Sankowski's recent O(n log log n) time planar algorithm as a subroutine.

Recently, there has been increasing interest and progress in improvising the approximation algorithm for well-known NP-Complete problems, particularly the approximation algorithm for the Vertex-Cover problem. Here we have proposed a polynomial time efficient algorithm for vertex-cover problem for more approximate to the optimal solution, which lead to the worst time complexity Θ(V 2 ) and space complexity Θ(V + E). We show that our proposed method is more approximate with example and theorem proof.

Journal of Combinatorial Theory, Series B, 2003

Let G be a 3-connected graph with n vertices on a non-spherical closed surface F 2 k of Euler genus k with sufficiently large representativity. In this paper, we first study a new cutting method which produces a spanning planar subgraph of G with a certain good property. This is used to show that such a graph G has a spanning 4-tree with at most maxf2k À 5; 0g vertices of degree 4. Using this result, we prove that for any integer t; if n is sufficiently large, then G has a connected subgraph with t vertices whose degree sum is at most 8t À 1: We also give a nearly sharp bound for the projective plane, torus and Klein bottle. Furthermore, using our cutting method, we prove that a 3-connected graph G on F 2 k with high representativity has a 3-walk in which at most maxf2k À 4; 0g vertices are visited three times, and an 8-covering with at most maxf4k À 8; 0g vertices of degree 7 or 8. Moreover, a 4-connected G has a 4-covering with at most maxf4k À 6; 0g vertices of degree 4.

Discrete & Computational Geometry, 2006

Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e > 4v edges is at least ce 3 /v 2 , where c > 0 is an absolute constant. This result, known as the "Crossing Lemma," has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c > 1024/31827 > 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v − 2); and (2) the crossing number of any graph is at least 7 3 e − 25 3 (v − 2). Both bounds are tight up to an additive constant (the latter one in the range 4v ≤ e ≤ 5v).

Proceedings of the twenty-third annual ACM symposium on Theory of computing - STOC '91, 1991

We give an algorithm for imbedding a graph G of n vertices onto an oriented surface of minimal genus g. If g > 0 then we also construct a forbidden subgraph of G which is homeomorphic to a graph of size exp(O(g)!) which cannot be imbedded on a surface of genus g-1. Our algorithm takes sequential time exp(O(g)!)n O(1) . Since exp(O(g)!) = exp(exp(O(glog(g)))), our algorithm is polynomial time for genus g=O(loglog(n)/logloglog(n)). A simple parallel implementation of our algorithm takes parallel time processors. We give also the smallest known upper bound, namely exp(O(g)!), on the number F(g) of homeomorphic distinct forbidden subgraphs for graph imbeddings onto a surface of genus g.

Theoretical Computer Science, 2013

Parameterised approximation is a relatively new but growing field of interest. It merges two ways of dealing with NP-hard optimisation problems, namely polynomial approximation and exact parameterised (exponential-time) algorithms. We exemplify this idea by designing and analysing parameterised approximation algorithms for minimum vertex cover. More specifically, we provide a simple algorithm that works on any approximation ratio of the form 2l+1 l+1 , l = 1, 2, . . ., and has complexity that outperforms previously published algorithms based on sophisticated exact parameterised algorithms.

Lecture Notes in Computer Science, 1995

We give a survey of recent techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general resuits or those results which have an algorithmic flavor, including the recent results of the authors.

Journal of Combinatorial Theory, Series B, 2001

A graph G is an apex graph if it contains a vertex w such that G − w is a planar graph. It is easy to see that the genus g(G) of the apex graph G is bounded above by τ − 1, where τ is the minimum face cover of the neighbors of w, taken over all planar embeddings of G − w. The main result of this paper is the linear lower bound g(G) ≥ τ /160 (if G − w is 3-connected and τ > 1). It is also proved that the minimum face cover problem is NP-hard for planar triangulations and that the minimum vertex cover is NP-hard for 2-connected cubic planar graphs. Finally, it is shown that computing the genus of apex graphs is NP-hard.

Information Processing Letters, 1989