Graph Withn Vertices Research Articles

Some geometric applications of Dilworth’s theorem

A geometric graph is a graph drawn in the plane such that its edges are closed line segments and no three vertices are collinear. We settle an old question of Avital, Hanani, Erd?s, Kupitz, and Perles by showing that every geometric graph withn vertices andm>k4n edges containsk+1 pairwise disjoint edges. We also prove that, given a set of pointsV and a set of axis-parallel rectangles in the plane, then either there arek+1 rectangles such that no point ofV belongs to more than one of them, or we can find an at most 2·105k8 element subset ofV meeting all rectangles. This improves a result of Ding, Seymour, and Winkler. Both proofs are based on Dilworth's theorem on partially ordered sets.

A new scaling algorithm for the maximum mean cut problem

We present a new scaling algorithm for the maximum mean cut problem. The mean of a cut is defined by the cut capacity divided by the number of arcs crossing the cut. The algorithm uses an approximate binary search and solves the circulation feasibility problem with relaxed capacity bounds. The maximum mean cut problem has recently been studied as a dual analogue of the minimum mean cycle problem in the framework of the minimum cost flow problem by Ervolina and McCormick. A networkN=(G, lower, upper) with lower and upper arc capacities is said to be ź-feasible ifN has a feasible circulation when we relax the capacity bounds by ź; that is, we use (lower(a)- ź, upper(a)+ź) bounds instead of (lower(a), upper(a)) bounds for each arca źA. During an approximate binary search we maintain two bounds,LB andUB, such thatN is LB-infeasible andUB-feasible, and we reduce the interval size (LB, UB) by at least one-third at each iteration. For a graph withn vertices, m arcs, and integer capacities bounded byU, the running time of this algorithm is O(mn log(nU). This time bound is better than the time achieved by McCormick and Ervolina under thesimilarity condition (that is,U=O(no(1))). Our algorithm can be naturally used for the circulation feasibility problem, and thus provides a new scaling algorithm for the minimum cut problem.

AnO(n 2) algorithm for finding the compact sets of a graph

A special clustering problem is discussed in this paper, called the compact set problem. The goal of the problem is to find all compact sets in a complete, weighted, undirected graph withn vertices. A subsetC of vertices is called a compact set if 1<|C|<n and the maximum weight among all edges inC is smaller than the minimum weight among all edges connecting one vertex inC and the other vertex not inC. An algorithm with complexityO(n2) is given for the problem improving the previous results.

A restricted crossbar implementation and its applications

In this article a new concept for permutation devices is shown. Such devices, like crossbars, are used within connection networks for parallel computers.The idea shown here is to partition the complete graph with n vertices into n &amp;minus; 1 different partitions and to switch very fast between these partitions, so that the user sees a complete graph structure. Finally a compact VLSI layout is described which can be easily implemented using automatic design tools.

An efficient distributed algorithm for maximum matching in general graphs

We present a distributed algorithm for maximum cardinality matching in general graphs. On a general graph withn vertices, our algorithm requiresO(n 5/2) messages in the worst case. On trees, our algorithm computes a maximum matching usingO(n) messages after the election of a leader.

Solving some combinatorial problems on arrays with one-way dataflow

In this paper we present algorithms for solving some combinatorial problems on one-dimensional processor arrays in which data flows in only one direction through the array. The problems we consider are: ranking the elements in a chain of sizen, rooting a spanning tree withn vertices, and computing biconnected components of a connected graph withn vertices. We show that each of these problems can be solved using arrays of sizen in which the data enters at the first cell and flows through the array in only one direction until it leaves the last cell as output. We also show how the biconnectivity algorithm for the array yields a new sequential algorithm for computing biconnected components which uses onlyO(n) locations of random access memory.

How to draw a planar graph on a grid

Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fary embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fary embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

A linear systolic algorithm for the connected component problem

A linear systolic algorithm is proposed for the connected component problem. Given an undirected graph withn vertices,m edges, andp connected components, the proposed algorithm uses (n −p + 1) cells pipelined together to figure out, in (m + 3(n −p) + 2) systolic cycles, which component each vertex belongs to.

Bounds of eigenvalues of a graph

LetG be a simple graph withn vertices. We denote by λi(G) thei-th largest eigenvalue ofG. In this paper, several results are presented concerning bounds on the eigenvalues ofG. In particular, it is shown that −1⩽λ2(G)⩽(n−2)/2, and the left hand equality holds if and only ifG is a complete graph with at least two vertices; the right hand equality holds if and only ifn is even andG℞2K n/2.

An adaptive and cost-optimal parallel algorithm for minimum spanning trees

A parallel algorithm is described for computing the minimum spanning tree of an undirected, connected and weighted graph withn vertices. We assume a shared-memory single-instruction-stream, multiple-data-stream model of computation which does not allow read or write conflicts. The algorithm is adaptive in the sense that it usesn1−e processors and runs inO(n1+e) time wheree lies between 0 and 1 and depends on the number of available processors. In view of the obvious Ω(n2) lower bound on the number of operations required to compute a minimum spanning tree, the algorithm is also cost-optimal.

Note on a Ramsey-Turán type problem

Solving a problem of Erdos the existence of a graph withn vertices,cn 2 edges and withoutK(4) andK(3, 3, 3) is proved. It remains an open question whether all such graphs containK(2, 2, 2).

Graph algorithms on a tree-structured parallel computer

Parallel algorithms for some graph-theoretic problems on a tree-structured computer are presented. In particular, ifp denotes the number of processing elements, algorithms that run inO(n2/p) time for finding connected components, transitive closure and the minimum spanning tree of an undirected graph withn vertices are obtained.

On factors in random graphs

The following result is proved: LetGn,p be a random graph withn vertices and probabilityp for an edge. Ifp is such that the random graph has min-degree at leastr with probability 1, then anyf-factor 1≦f≦r exists with probability 1, asn→∞.

Computing connected components on parallel computers

We present a parallel algorithm which uses n 2 processors to find the connected components of an undirected graph with n vertices in time O (log 2 n ). An O (log 2 n ) time bound also can be achieved using only n ⌈ n /⌈log 2 n ⌉⌉ processors. The algorithm can be used to find the transitive closure of a symmetric Boolean matrix. We assume that the processors have access to a common memory. Simultaneous access to the same location is permitted for fetch instructions but not for store instructions.

Space bounds for a game on graphs

We study a one-person game played by placing pebbles, according to certain rules, on the vertices of a directed graph. In [3] it was shown that for each graph withn vertices and maximum in-degreed, there is a pebbling strategy which requires at mostc(d) n/logn pebbles. Here we show that this bound is tight to within a constant factor. We also analyze a variety of pebbling algorithms, including one which achieves the 0(n/logn) bound.

A search strategy for the elementary cycles of a directed graph

The most successful known algorithms enumerating the elementary cycles of a directed graph are based on a backtracking strategy. Such existing algorithms are discussed and a new backtracking algorithm is proposed which is bounded byO(N +M(C + 1)) time, for a directed graph withN vertices,M edges andC elementary cycles.

Generation of Rosary permutations expressed in Hamiltonian circuits

Systematic generation of a specific class of permutations fundamental to scheduling problems is described. In a nonoriented complete graph with n vertices, Hamiltonian circuits equivalent to 1/2( n - 1)! specific permutations of n elements, termed rosary permutations, can be defined. Each of them corresponds to two circular permutations which mirror-image each other, and is generated successively by a number system covering 3·4· ··· ·( n - 1) sets of edges. Every set of edges { e k }, 1 ≤ e k ≤ k , 3 ≤ k ≤ n - 1 is determined recursively by constructing a Hamiltonian circuit with k vertices from a Hamiltonian circuit with k - 1 vertices, starting with the Hamiltonian circuit of 3 vertices. The basic operation consists of transposition of a pair of adjacent vertices where the position of the pair in the permutation is determined by { e k }. Two algorithms treating the same example for five vertices are presented. It is very easy to derive all possible n ! permutations from the 1/2( n - 1)! rosary permutations by cycling the permutations and by taking them in the reverse order—procedures which can be performed fairly efficiently by computer.