Abstract not found

A linear arrangement of an n-vertex graph is a one-to-one mapping of its vertices to the integers f1; : : : ; ng. The bandwidth of a linear arrangement is the maximum difference between mapped values of adjacent vertices. The problem of finding a linear arrangement with smallest possible bandwidth in NP-hard. We present a randomized algorithm that runs in nearly linear time and outputs a linear arrangement whose bandwidth is within a polylogarithmic multiplicative factor of optimal. Our algorithm is based on a new notion, called volume respecting embeddings, which is a natural extension of small distortion embeddings of Bourgain and of Linial, London and Rabinovich. 1 Introduction We consider the problem of minimizing the bandwidth of an undirected connected graph G(V; E), where n = jV j and m = jEj. One needs to find a linear arrangement of the vertices, namely, a one-to-one mapping f : V \Gamma! f1; 2; : : : ng, for which the bandwidth, i.e. max (i;j)2E jf(i) \Gamma f(j)j, i...

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fill-in exist, we can list their potential maximal cliques in polynomial time. Our approach unies these algorithms. Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs, for which the treewidth and the minimum fill-in problems were open.

We present simple semi-definite programming relaxations for the NP-hard minimum bandwidth and minimum length linear ordering problems. We then show how these relaxations can be rounded in a natural way (via random projection) to obtain approximation guarantees for both of these vertex-ordering problems.

Traditionally, efficient algorithms for computing a minimal triangulation of a graph (i.e. embedding a graph into a triangulated graph by adding an inclusion-minimal set of edges) required first computing a special ordering on the vertices of the graph, called a minimal ordering. We give a new algorithm which efficiently computes a minimal triangulation using an arbitrary ordering on the vertices. 1 Introduction. Computing a minimal triangulation consists in embedding a given graph into a triangulated graph by adding a set of edges (called a fill). If the set of edges added is inclusion-minimal, the fill is said to be minimal, and the corresponding triangulated graph is called a minimal triangulation. Finding a fill that is minimum is NP-complete ([10]). Given a graph G and any ordering ff on its vertices, an associated fill can be computed by repeatedly choosing the next vertex x in order ff, adding the edges necessary to make the neighborhood of x into a clique (i.e. by making x si...

A spanning tree of a graph is a k-additive tree spanner whenever the distance of every two vertices in the graph or in the tree differs by at most k. In this paper we show that certain classes of graphs, as distance-hereditary graphs, interval graphs, asteroidal-triple free graphs, allow some constant k such that every member of the class has some k-additive tree spanner. On the other hand, there are chordal graphs without k-additive tree spanner for arbitrary large k.

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In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in AT-free graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT(x, y) ≤ dG(x, y) + r. Among other results, we show that AT-free graphs have a system of two collective additive tree 2-spanners (whereas there are trapezoid graphs that do not admit any additive tree 2-spanner). Furthermore, based on this collection, we derive a compact and efficient routing scheme. Also, any DSP-graph (there exists a dominating shortest path) admits an additive tree 4-spanner, a system of two collective additive tree 3-spanners and a system of five collective additive tree 2-spanners.

Fishburn, Tanenbaum and Trenk [4] de ne the linear discrepancy ld(P ) of a poset P = (V; < P ) as the minimum integer k 0 for which there exists a bijection f : V ! f1; 2; : : : ; jV jg such that u < P v implies f(u) < f(v) and ujj P v implies jf(u) f(v)j k. In [5] they prove that the linear discrepancy of a poset equals the bandwidth of its cocomparability graph.

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