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Pathwidth
About: Pathwidth is a research topic. Over the lifetime, 8354 publications have been published within this topic receiving 252234 citations.
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TL;DR: In this paper, it is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph.
Abstract: The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...
1,762 citations
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01 Jan 1962
TL;DR: In this article, the axiom of choice of choice is used to define connectedness path problems in directed graphs and cyclic graphs, as well as Galois correspondences of connectedness paths.
Abstract: Fundamental concepts Connectedness Path problems Trees Leaves and lobes The axiom of choice Matching theorems Directed graphs Acyclic graphs Partial order Binary relations and Galois correspondences Connecting paths Dominating sets, covering sets and independent sets Chromatic graphs Groups and graphs Bibliography List of concepts Index of names.
1,732 citations
TL;DR: Every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm that determines whether the treewidth of G is at most at most some constant $k$ and finds a tree-decomposition of G withtreewidth at most k.
Abstract: In this paper, we give for constant $k$ a linear-time algorithm that, given a graph $G=(V,E)$, determines whether the treewidth of $G$ is at most $k$ and, if so, finds a tree-decomposition of $G$ with treewidth at most $k$. A consequence is that every minor-closed class of graphs that does not contain all planar graphs has a linear-time recognition algorithm. Another consequence is that a similar result holds when we look instead for path-decompositions with pathwidth at most some constant $k$.
1,666 citations
TL;DR: The consecutive ones test for the consecutive ones property in matrices and for graph planarity is extended to a test for interval graphs using a recently discovered fast recognition algorithm for chordal graphs.
1,622 citations
TL;DR: A family of efficient kernels for large graphs with discrete node labels based on the Weisfeiler-Lehman test of isomorphism on graphs that outperform state-of-the-art graph kernels on several graph classification benchmark data sets in terms of accuracy and runtime.
Abstract: In this article, we propose a family of efficient kernels for large graphs with discrete node labels. Key to our method is a rapid feature extraction scheme based on the Weisfeiler-Lehman test of isomorphism on graphs. It maps the original graph to a sequence of graphs, whose node attributes capture topological and label information. A family of kernels can be defined based on this Weisfeiler-Lehman sequence of graphs, including a highly efficient kernel comparing subtree-like patterns. Its runtime scales only linearly in the number of edges of the graphs and the length of the Weisfeiler-Lehman graph sequence. In our experimental evaluation, our kernels outperform state-of-the-art graph kernels on several graph classification benchmark data sets in terms of accuracy and runtime. Our kernels open the door to large-scale applications of graph kernels in various disciplines such as computational biology and social network analysis.
1,552 citations