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79
A survey of the theory of hypercube graphs
 Computers and Mathematics with Applications 15
, 1988
"... AlmtractWe present acomprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the ncube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced ..."
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Cited by 62 (1 self)
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AlmtractWe present acomprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the ncube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced paths and cycles. Finally, various embedding and packing problems are discussed, including the determination of the cubical dimension of a given cubical graph. I.
Shallow Excluded Minors and Improved Graph Decompositions
, 1994
"... In this paper we introduce the notion of the limiteddepth minor exclusion and show that graphs that exclude small limiteddepth minors have relatively small separators. In particular, we prove that for any graph that excludes K h as a depth l minor, we can find a separator of size O(lh 2 log n n=l) ..."
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Cited by 50 (2 self)
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In this paper we introduce the notion of the limiteddepth minor exclusion and show that graphs that exclude small limiteddepth minors have relatively small separators. In particular, we prove that for any graph that excludes K h as a depth l minor, we can find a separator of size O(lh 2 log n n=l). This, in turn, implies that any graph that excludes K h as a minor has an O(h p n log n)sized separator, improving the result of Alon, Seymour, and Thomas for the case where h AE p log n. We show that the ddimensional simplicial graphs with constant aspect ratio, defined by Miller and Thurston, exclude K h minors of depth L for h = \Omega\Gamma L d\Gamma1 ) when d is a constant. These graphs arise in finite element computations. Our proof of separator existence is constructive and gives an algorithm to find the tcutcovers decomposition, introduced by Kaklamanis, Krizanc, and Rao, in graphs that exclude small depth minors. This has two interesting implications. F...
Consensus Algorithms for the Generation of All Maximal Bicliques
, 2002
"... We describe a new algorithm for generating all maximal bicliques (i.e. complete bipartite, not necessarily induced subgraphs) of a graph. The algorithm is inspired by, and is quite similar to, the consensus method used in propositional logic. We show that some variants of the algorithm are totally p ..."
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Cited by 44 (5 self)
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We describe a new algorithm for generating all maximal bicliques (i.e. complete bipartite, not necessarily induced subgraphs) of a graph. The algorithm is inspired by, and is quite similar to, the consensus method used in propositional logic. We show that some variants of the algorithm are totally polynomial, and even incrementally polynomial. The total complexity of the most efficient variant of the algorithms presented here is polynomial in the input size, and only linear in the output size. Computational experiments demonstrate its high efficiency on randomly generated graphs with up to 2,000 vertices and 20,000 edges.
Distance realization problems with applications to Internet tomography
"... In recent years, a variety of graph optimization problems have arisen in which the graphs involved are much too large for the usual algorithms to be effective. In these cases, even though we are not able to examine the entire graph (which may be changing dynamically), we would still like to deduce v ..."
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Cited by 21 (2 self)
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In recent years, a variety of graph optimization problems have arisen in which the graphs involved are much too large for the usual algorithms to be effective. In these cases, even though we are not able to examine the entire graph (which may be changing dynamically), we would still like to deduce various properties of it, such as the size of a connected component, the set of neighbors of a subset of vertices, etc. In this paper, we study a class of problems, called distance realization problems, which arise in the study of Internet data traffic models. uppose we are given a set S of terminal nodes, taken from some (unknown) weighted graph. A basic problem is to reconstruct a weighted graph G including S with possibly additional vertices, that realizes the given distance matrix for S. We will first show that this problem is not only difficult but the solution is often unstable in the sense that even if all distances between nodes in S decrease, the solution can increase by a factor proport...
On universal graphs for spanning trees
 Proc. London Math. Soc
, 1983
"... A number of papers [1,2,3,4,6] recently have been concerned with the following question. What is the minimum number s{n) of edges a graph G on n vertices can have so that any tree on n vertices is isomorphic to some spanning tree of G? We call such a graph universal for spanning trees. Since Kn, the ..."
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Cited by 18 (4 self)
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A number of papers [1,2,3,4,6] recently have been concerned with the following question. What is the minimum number s{n) of edges a graph G on n vertices can have so that any tree on n vertices is isomorphic to some spanning tree of G? We call such a graph universal for spanning trees. Since Kn, the complete graph
Small InducedUniversal Graphs and Compact Implicit Graph Representations
 In Proc. 43’rd annual IEEE Symp. on Foundations of Computer Science
, 2002
"... We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound ..."
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Cited by 18 (1 self)
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We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound of the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted trees.
Sparse universal graphs for boundeddegree graphs
"... Let H be a family of graphs. A graph T is Huniversal if it contains a copy of each H ∈ H as a subgraph. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. 2 2 − For all positive integers k and n, we construct an H(k, n)universal graph T with Ok(n k log 4 k n) edge ..."
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Cited by 15 (2 self)
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Let H be a family of graphs. A graph T is Huniversal if it contains a copy of each H ∈ H as a subgraph. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. 2 2 − For all positive integers k and n, we construct an H(k, n)universal graph T with Ok(n k log 4 k n) edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n2−2/k) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic, and is based on a novel graph decomposition result and on the properties of random walks on expanders. 1
Introduction to papers on the modeling and analysis of network data
 Ann. Appl. Statist
, 2010
"... ar ..."