Results 1  10
of
52
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) ..."
Abstract

Cited by 36 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 26 (4 self)
 Add to MetaCart
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.
The intrinsic dimensionality of graphs
 In STOC
, 2003
"... Abstract. We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [16]. For a graph G, let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of Z d ∞, the infinite graph with vertex set Z d and an edge (u, v) whenever u − v ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
Abstract. We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [16]. For a graph G, let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of Z d ∞, the infinite graph with vertex set Z d and an edge (u, v) whenever u − v ∞ = 1. The growth rate of G, denoted ρG, is the minimum ρ such that every ball of radius r> 1 in G contains at most r ρ vertices. By simple volume arguments, dim(G) = Ω(ρG). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρG) for every graph G. Previously, it was unknown whether dim(G) could be bounded above by any function of ρG. We show that a weaker form of Levin’s conjecture holds by proving that dim(G) = O(ρG log ρG) for any graph G. We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for which dim(G) = Ω(ρG log ρG). For several special families of graphs (e.g., planar graphs), we salvage the strong form, showing that dim(G) = O(ρG). Our results extend to a variant of the conjecture for finitedimensional Euclidean spaces posed by Linial [15] and independently by Benjamini and Schramm [22]. 1.
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 ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
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.
Nearoptimal universal graphs for graphs with bounded degrees
 APPROXRANDOM 2001, LNCS 3139 (2001) 170180 the electronic journal of combinatorics 9
, 2001
"... Abstract. Let H be a family of graphs. We say that G is Huniversal if, for each H ∈H,the graph G contains a subgraph isomorphic to H. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an H(k, ..."
Abstract

Cited by 10 (9 self)
 Add to MetaCart
Abstract. Let H be a family of graphs. We say that G is Huniversal if, for each H ∈H,the graph G contains a subgraph isomorphic to H. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an H(k, n)universal graph Γ (k, n) with O(n 2−2/k (log n) 1+8/k) edges. This is optimal up to a small polylogarithmic factor, as Ω(n 2−2/k) is a lower bound for the number of edges in any such graph. En route, we use the probabilistic method in a rather unusual way. After presenting a deterministic construction of the graph Γ (k, n), we prove, using a probabilistic argument, that Γ (k, n) isH(k, n)universal. So we use the probabilistic method to prove that an explicit construction satisfies certain properties, rather than showing the existence of a construction that satisfies these properties. 1Introduction and Main Result For a family H of graphs, a graph G is Huniversal if, for each H ∈H,the
Embeddings in Hypercubes
, 1988
"... : One important aspect of efficient use of a hypercube computer to solve a given problem is the assignment of subtasks to processors in such a way that the communication overhead is low. The subtasks and their intercommunication requirements can be modeled by a graph, and the assignment of subtasks ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
: One important aspect of efficient use of a hypercube computer to solve a given problem is the assignment of subtasks to processors in such a way that the communication overhead is low. The subtasks and their intercommunication requirements can be modeled by a graph, and the assignment of subtasks to processors viewed as an embedding of the task graph into the graph of the hypercube network. We survey the known results concerning such embeddings, including expansion /dilation tradeoffs for general graphs, embeddings of meshes and trees, packings of multiple copies of a graph, the complexity of finding good embeddings, and critical graphs which are minimal with respect to some property. In addition, we describe several open problems. Keywords: hypercube computer, ncube, embedding, dilation, expansion, cubical, packing, random graphs, critical graphs. 1 Introduction Let Q n denote an ndimensional binary cube where the nodes of Q n are all the binary n tuples and two nodes are ad...
Optimal Broadcasting in Hypercubes with Dynamic Faults
 INFORMATION PROCESSING LETTERS
, 1999
"... We consider the broadcasting problem in the shouting communication mode in which any node of a network can inform all its neighbours in one time step. In addition, during any time step a number of links less than the edgeconnectivity of the network can be faulty. The problem is to find an upper bou ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
We consider the broadcasting problem in the shouting communication mode in which any node of a network can inform all its neighbours in one time step. In addition, during any time step a number of links less than the edgeconnectivity of the network can be faulty. The problem is to find an upper bound on the number of time steps necessary to complete broadcasting under this additional assumption. Fraigniaud and Peyrat proved for the ndimensional hypercube that n + O(log n) time steps are sufficient. De Marco and Vaccaro decreased the upper bound to n+7 and showed a worst case lower bound n + 2 for n 3. We prove that n + 2 time steps are sufficient. Our method is related to the isoperimetric problem in graphs and can be applied to other networks.