Realizing degree sequences in parallel

A sequence $d$ of integers is a degree sequence if there exists a (simple) graph $G$ such that the components of $d$ are equal to the degrees of the vertices of $G$. The graph $G$ is said to be a realization of $d$. We provide an efficient parallel algorithm to realize $d$. Before our result, it was not known if the problem of realizing $d$ is in $NC$

Parallel enumeration of degree sequences of simple graphs. II.

Abstract In the paper we report on the parallel enumeration of the degree sequences (their number is denoted by G(n)) and zerofree degree sequences (their number is denoted by (Gz(n)) of simple graphs on n = 30 and n = 31 vertices. Among others we obtained that the number of zerofree degree sequences of graphs on n = 30 vertices is Gz(30) = 5 876 236 938 019 300 and on n = 31 vertices is Gz(31) = 22 974 847 474 172 374. Due to Corollary 21 in [52] these results give the number of degree sequences of simple graphs on 30 and 31 vertices.</jats:p

Deciding football sequences

An open problem posed by the first author is the complexity to decide whether a sequence of nonnegative integer numbers can be the final score of a football tournament. In this paper we propose polynomial time approximate and exponential time exact algorithms which solve the problem

On the parallel complexity of degree sequence problems

We describe a robust and efficient implementation of the Bentley-Ottmann sweep line algorithm based on the LEDA library of efficient data types and algorithms. The program computes the planar graph $G$ induced by a set $S$ of straight line segments in the plane. The nodes of $G$ are all endpoints and all proper intersection points of segments in $S$. The edges of $G$ are the maximal relatively open subsegments of segments in $S$ that contain no node of $G$. All edges are directed from left to right or upwards. The algorithm runs in time $O((n+s) log n)$ where $n$ is the number of segments and $s$ is the number of vertices of the graph $G$. The implementation uses exact arithmetic for the reliable realization of the geometric primitives and it uses floating point filters to reduce the overhead of exact arithmetic

Realizing degree sequences in parallel

A sequence d of integers is a degree sequence if there exists a (simple) graph G such that the components of d are equal to the degrees of the vertices of G. The graph G is said to be a realization of d. We provide an efficient parallel algorithm to realize d; the algorithm runs in O(log n) me using O(n + m) CRCW PRAM processors, where n and m are the number of vertices and edges G. Before our result, it was not known if the problem of realizing d is in NC