scispace - formally typeset

Graph power

About: Graph power is a(n) research topic. Over the lifetime, 7339 publication(s) have been published within this topic receiving 156281 citation(s). more


Open accessJournal ArticleDOI: 10.1016/0304-3975(76)90059-1
Abstract: It is widely believed that showing a problem to be NP -complete is tantamount to proving its computational intractability. In this paper we show that a number of NP -complete problems remain NP -complete even when their domains are substantially restricted. First we show the completeness of Simple Max Cut (Max Cut with edge weights restricted to value 1), and, as a corollary, the completeness of the Optimal Linear Arrangement problem. We then show that even if the domains of the Node Cover and Directed Hamiltonian Path problems are restricted to planar graphs, the two problems remain NP -complete, and that these and other graph problems remain NP -complete even when their domains are restricted to graphs with low node degrees. For Graph 3-Colorability, Node Cover, and Undirected Hamiltonian Circuit, we determine essentially the lowest possible upper bounds on node degree for which the problems remain NP -complete. more

Topics: Maximum cut (60%), Forbidden graph characterization (59%), Hamiltonian path problem (58%) more

2,062 Citations

Open accessJournal ArticleDOI: 10.1016/J.ACHA.2010.04.005
Abstract: We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator Ttg = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains. more

Topics: Line graph (65%), Spectral graph theory (64%), Wavelet (64%) more

1,395 Citations

Open accessJournal ArticleDOI: 10.1006/JCTB.1995.1006
Neil Robertson1, Paul Seymour1Institutions (1)
Abstract: We describe an algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G , decide if there are k mutually vertex-disjoint paths of G joining the pairs. more

Topics: Bound graph (65%), Graph power (65%), k-vertex-connected graph (63%) more

1,304 Citations

Journal ArticleDOI: 10.1109/TKDE.2007.46
Abstract: This work presents a new perspective on characterizing the similarity between elements of a database or, more generally, nodes of a weighted and undirected graph. It is based on a Markov-chain model of random walk through the database. More precisely, we compute quantities (the average commute time, the pseudoinverse of the Laplacian matrix of the graph, etc.) that provide similarities between any pair of nodes, having the nice property of increasing when the number of paths connecting those elements increases and when the "length" of paths decreases. It turns out that the square root of the average commute time is a Euclidean distance and that the pseudoinverse of the Laplacian matrix is a kernel matrix (its elements are inner products closely related to commute times). A principal component analysis (PCA) of the graph is introduced for computing the subspace projection of the node vectors in a manner that preserves as much variance as possible in terms of the Euclidean commute-time distance. This graph PCA provides a nice interpretation to the "Fiedler vector," widely used for graph partitioning. The model is evaluated on a collaborative-recommendation task where suggestions are made about which movies people should watch based upon what they watched in the past. Experimental results on the MovieLens database show that the Laplacian-based similarities perform well in comparison with other methods. The model, which nicely fits into the so-called "statistical relational learning" framework, could also be used to compute document or word similarities, and, more generally, it could be applied to machine-learning and pattern-recognition tasks involving a relational database more

Topics: Laplacian matrix (66%), Null graph (63%), Random geometric graph (63%) more

1,139 Citations

Proceedings ArticleDOI: 10.1145/224170.224228
Bruce Hendrickson1, Robert W. Leland1Institutions (1)
08 Dec 1995-
Abstract: The graph partitioning problem is that of dividing the vertices of a graph into sets of specified sizes such that few edges cross between sets. This NP-complete problem arises in many important scientific and engineering problems. Prominent examples include the decomposition of data structures for parallel computation, the placement of circuit elements and the ordering of sparse matrix computations. We present a multilevel algorithm for graph partitioning in which the graph is approximated by a sequence of increasingly smaller graphs. The smallest graph is then partitioned using a spectral method, and this partition is propagated back through the hierarchy of graphs. A variant of the Kernighan-Lin algorithm is applied periodically to refine the partition. The entire algorithm can be implemented to execute in time proportional to the size of the original graph. Experiments indicate that, relative to other advanced methods, the multilevel algorithm produces high quality partitions at low cost. more

Topics: Graph partition (75%), Strength of a graph (71%), Modular decomposition (71%) more

1,122 Citations

No. of papers in the topic in previous years

Top Attributes

Show by:

Topic's top 5 most impactful authors

Noga Alon

32 papers, 1K citations

Daniël Paulusma

31 papers, 511 citations

Ivan Gutman

23 papers, 1K citations

Ralph J. Faudree

21 papers, 376 citations

Petr A. Golovach

19 papers, 328 citations

Network Information
Related Topics (5)
Indifference graph

10.8K papers, 287.9K citations

95% related
1-planar graph

6.2K papers, 166.4K citations

94% related

8.3K papers, 252.2K citations

94% related
Complement graph

6.4K papers, 166.5K citations

94% related
Chordal graph

12.8K papers, 314.2K citations

93% related