Topic

# Pathwidth

About: Pathwidth is a(n) research topic. Over the lifetime, 8354 publication(s) have been published within this topic receiving 252234 citation(s).

...read more

##### Papers

More filters

••

Abstract: In this paper we present a new data structure for representing Boolean functions and an associated set of manipulation algorithms. Functions are represented by directed, acyclic graphs in a manner similar to the representations introduced by Lee [1] and Akers [2], but with further restrictions on the ordering of decision variables in the graph. Although a function requires, in the worst case, a graph of size exponential in the number of arguments, many of the functions encountered in typical applications have a more reasonable representation. Our algorithms have time complexity proportional to the sizes of the graphs being operated on, and hence are quite efficient as long as the graphs do not grow too large. We present experimental results from applying these algorithms to problems in logic design verification that demonstrate the practicality of our approach.

...read more

8,721 citations

•

01 Jan 2009-

TL;DR: The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial.

...read more

Abstract: Graphs.- Groups.- Transitive Graphs.- Arc-Transitive Graphs.- Generalized Polygons and Moore Graphs.- Homomorphisms.- Kneser Graphs.- Matrix Theory.- Interlacing.- Strongly Regular Graphs.- Two-Graphs.- Line Graphs and Eigenvalues.- The Laplacian of a Graph.- Cuts and Flows.- The Rank Polynomial.- Knots.- Knots and Eulerian Cycles.- Glossary of Symbols.- Index.

...read more

7,743 citations

•

14 Nov 1995-

Abstract: 1. Fundamental Concepts. Definitions and examples. Paths and proofs. Vertex degrees and counting. Degrees and algorithmic proof. 2. Trees and Distance. Basic properties. Spanning trees and enumeration. Optimization and trees. Eulerian graphs and digraphs. 3. Matchings and Factors. Matchings in bipartite graphs. Applications and algorithms. Matchings in general graphs. 4. Connectivity and Paths. Cuts and connectivity. k-connected graphs. Network flow problems. 5. Graph Coloring. Vertex colorings and upper bounds. Structure of k-chromatic graphs. Enumerative aspects. 6. Edges and Cycles. Line graphs and edge-coloring. Hamiltonian cycles. Complexity. 7. Planar Graphs. Embeddings and Eulers formula. Characterization of planar graphs. Parameters of planarity. 8. Additional Topics. Perfect graphs. Matroids. Ramsey theory. More extremal problems. Random graphs. Eigenvalues of graphs. Glossary of Terms. Glossary of Notation. References. Author Index. Subject Index.

...read more

7,114 citations

••

TL;DR: This work presents a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of theSize of the final partition obtained after multilevel refinement, and presents a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening.

...read more

Abstract: Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. of the 6th SIAM Conference on Parallel Processing for Scientific Computing, 1993, 445--452; Hendrickson and Leland, A Multilevel Algorithm for Partitioning Graphs, Tech. report SAND 93-1301, Sandia National Laboratories, Albuquerque, NM, 1993]. From the early work it was clear that multilevel techniques held great promise; however, it was not known if they can be made to consistently produce high quality partitions for graphs arising in a wide range of application domains. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme produces partitions that are consistently better than those produced by spectral partitioning schemes in substantially smaller time. Also, when our scheme is used to compute fill-reducing orderings for sparse matrices, it produces orderings that have substantially smaller fill than the widely used multiple minimum degree algorithm.

...read more

5,117 citations

12

••

Mark Newman

^{1}, Mark Newman^{2}, Steven H. Strogatz^{2}, Duncan J. Watts^{3}+1 more•Institutions (3)TL;DR: It is demonstrated that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

...read more

Abstract: Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the worldwide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

...read more

3,442 citations