Showing papers on "K-tree published in 1994"

Can visibility graphs Be represented compactly

Abstract: We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graphG, a familyG={G1,G2,...,Gk} is called aclique cover ofG if (i) eachGi is a clique or a bipartite clique, and (ii) the union ofGi isG. The size of the clique coverG is defined as ?i=1kni, whereni is the number of vertices inGi. Our main result is that there are visibility graphs ofn nonintersecting line segments in the plane whose smallest clique cover has size Ω(n2/log2n). An upper bound ofO(n2/logn) on the clique cover follows from a well-known result in extremal graph theory. On the other hand, we show that the visibility graph of a simple polygon always admits a clique cover of sizeO(nlog3n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn).

Parallel maximal cliques algorithms for interval graphs with applications

Abstract: In this paper, an O(n log n) time algorithm for finding all the maximal cliques of an interval graph is proposed. This algorithm can also be implemented in parallel in O(log n) time using O(n/sup 2/) processors. The maximal cliques of an interval graph contain important structural information. Many problems on interval graphs can be solved after all the maximal cliques are known. It is shown that cut vertices, bridges, and vertex connectivities can all be determined easily after the maximal cliques are known. Finally, the all-pair shortest path problem for interval graphs is solved based on the relationship between maximal cliques. The all-pair shortest path algorithm can also be parallelized in O(log n) time using O(n/sup 2/) processors. >

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

Abstract: A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn2/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mosttp-1( n ) cliques. Heretp-1( n ) is the number of edges in the Turan graph of ordern, which has no complete subgraphs of orderp.

Greedy maximum-clique decompositions

Abstract: A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. We have recently shown that any greedy clique decomposition of a graph of ordern has at mostn2/4 cliques. A greedy max-clique decomposition is a particular kind cf greedy clique decomposition where maximum cliques are removed, instead of just maximal ones. In this paper, we show that any greedy max-clique decompositionC of a graph of ordern has , wheren(C) is the number of vertices inC.

Parallel algorithms for k-tree recognition and its applications

Abstract: The class of k-trees generalize the notion of trees, maximal outer-planar graphs, and caterpillars. We present new parallel algorithms to recognize k-trees and to find a collection of k vertex disjoint paths between a specified vertex pair (u,v), called a uv-cable, in a k-tree. A parallel algorithm to compute the k paths in a k-tree is also presented. The algorithms are based on parallel construction of a directed graph as a representation for R-trees, referred to as underlying trees in this paper. The model of parallel computation used is the CRCW PRAM (concurrent read, concurrent write, parallel RAM), where more than one processor can concurrently read from or write into the same memory location during the same memory cycle. Writing conflicts are resolved in a nondeterministic fashion. The recognition algorithm runs in O(log/sup 2/ n) time using O(m+n) processors. Constructing a spatial graph and the underlying tree representations of the k-tree for fixed k takes only O(log n) time using O(n) processors. The parallel algorithms for a uv-cable and k paths take O(log n) time using O(n) processors if the underlying tree structure of the k-tree is available. >

Parallel Recognition and Location Algorithms for Chordal Graphs Using Distance Matrices

Abstract: We present efficient parallel algorithms for recognizing chordal graphs and locating all maximal cliques of a chordal graph G=(V,E). Our techniques are based on partitioning the vertex set V using information contained in the distance matrix of the graph. We use these properties to formulate parallel algorithms which, given a graph G=(V,E) and its adjacency-level sets, decide whether or not G is a chordal graph, and, if so, locate all maximal cliques of the graph in time O(k) by using δ2·n2/k processors on a CRCW-PRAM, where δ is the maximum degree of a vertex in G and 1≤k≤n. The construction of the adjacency-level sets can be done by computing first the distance matrix of the graph, in time O(logn) with O(nβ+DG) processors, where DG is the output size of the partitions and β=2.376, and then extracting all necessary set information. Hence, the overall time and processor complexity of both algorithms are O(logn) and O(max{δ2·n2/logn, nβ+DG}), respectively. These results imply that, for δ≤√nlogn, the proposed algorithms improve in performance upon the best-known algorithms for these problems.

A maximum clique derivation algorithm for simplification of incompletely specified machines

Abstract: In this paper, derivation problems of maximum cliques are discussed, which take place in simplification processes of incompletely specified machines. In many cases, graphs consisting of a lot of edges are generated in the processes. Thus, for many combinations of the edges, it should be examined whether a clique is made of them. Therefore, it is impossible to derive the maximum cliques within a reasonable time. This paper presents a new algorithm to derive all the maximum cliques, which is more suitable for the derivation of maximum cliques of the undirected graphs generated in the simplification processes. >

Parallel Algorithms for k-tree Recognition and its Applications

Abstract: The class of k-trees generalize the notion of trees, maximal outer-planar graphs, and caterpillars In this paper we present new parallel algorithms to recognize k-trees and to find a collection of k vertex disjoint paths between specified vertex pair (U, w), called a uvcable, in a k-tree A parallel algorithm to compute kpath in a k-tree is also presented The algorithms are based on parallel construction of a directed graph as a representation for k-trecs, referred to as underlying tree in this paper The model of parallel computation used is the CRCW PRAM (Concurrent Read Concurrent Write Parallel RAM), where more than one processor can concurrently read from or write into the same memory location during the same memory cycle Writing conflicts are resolved in a non-deterministic fashion The recognition algorithm runs in O(log2n) time using O(m + n) processors Constructing spatial graph and underlying tree representations of the k-tree for fixed k take only O(1og n) time using O(n) processors The parallel algorithms for uv-cable and k-path take O(1og n) time using O(n) processors if the underlying tree structure of the k-tree is available