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In this paper we introduce the notion of the limited-depth minor exclusion and show that graphs that exclude small limited-depth 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 d-dimensional 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 t-cut-covers decomposition, introduced by Kaklamanis, Krizanc, and Rao, in graphs that exclude small depth minors. This has two interesting implications. F...

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.

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

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 finite-dimensional Euclidean spaces posed by Linial [15] and independently by Benjamini and Schramm [22]. 1.

We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a node-induced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all n-node graphs of fixed arboricity k as node-induced 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.

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...

Abstract. Let H be a family of graphs. We say that G is H-universal 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 H-universal if, for each H ∈H,the

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 edge-connectivity 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 n-dimensional 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.

We show that the edges of a graph with maximum edge density d can always be oriented such that each vertex has in-degree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, in-degree 3 is achieved for planar graphs. This immediately gives a space-optimal data structure that answers edge membership queries in a maximum edge density-d graph in O(log d) time. Keywords Graph orientation, edge density, Hall condition, balanced adjacency lists, edge membership queries 1 The Theorem Let G be an undirected graph with n vertices and m edges. The parameter ffi(G) = m n is commonly called the edge density of G. The maximum (edge) density is the smallest integer d such that the edge density of any subgraph of G does not exceed d. More precisely, d = dmaxfffi(G 0 ) j G 0 is a subgraph of Gge. For example, d 1 for trees, d 3 for planar graphs, d = d 1 2 log 2 ne for hypercubes [GG,AH], and d d 1 2 (c \Gamma...