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Feedback vertex set
About: Feedback vertex set is a research topic. Over the lifetime, 2205 publications have been published within this topic receiving 49218 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
TL;DR: In this article, the generalized vertex median of a weighted graph may be found by complete enumeration or by some heuristic method, and a method that seems to perform well in comparison with others found in the literature is proposed.
Abstract: The generalized vertex median of a weighted graph may be found by complete enumeration or by some heuristic method. This paper investigates alternatives and proposes a method that seems to perform well in comparison with others found in the literature.
802 citations
TL;DR: This work shows that INDEPENDENT SET is complete for W, and the W Hierarchy of parameterized problems was defined, and complete problems were identified for the classes W [ t ] for t ⩾ 2.
659 citations
TL;DR: The motivation in the present work is to "assign" this Laplacian eigenvalue when relative positions of various elements dictate the interconnection of the underlying weighted graph, so as to "synthesize" information graphs that have desirable system theoretic properties.
Abstract: We consider the set G consisting of graphs of fixed order and weighted edges. The vertex set of graphs in G will correspond to point masses and the weight for an edge between two vertices is a functional of the distance between them. We pose the problem of finding the best vertex positional configuration in the presence of an additional proximity constraint, in the sense that, the second smallest eigenvalue of the corresponding graph Laplacian is maximized. In many recent applications of algebraic graph theory in systems and control, the second smallest eigenvalue of Laplacian has emerged as a critical parameter that influences the stability and robustness properties of dynamic systems that operate over an information network. Our motivation in the present work is to "assign" this Laplacian eigenvalue when relative positions of various elements dictate the interconnection of the underlying weighted graph. In this venue, one would then be able to "synthesize" information graphs that have desirable system theoretic properties.
605 citations
TL;DR: The vertex method can avoid abnormality due to the discretization technique on the variables domain and the widening of the resulting function value set due to multi-occurrence of variables in the functional expression by conventional interval analysis methods.
550 citations