Let G = (V, E) be a directed graph and n denote |V|. We show that G is k-vertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k- I)-dimensional space Rk-l, ~ : V ~Rk-l, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91-102. Using this characterization, a directed graph can be tested for k-vertex connectivity by a Monte Carlo algo-rithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm in ex-pected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algo-rithm improves on the best previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both al-gorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (logn) times the respective sequential time complexi-ties. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of s-t numberings, we give a combinatorial construction of a directed s-t nulmberiug for any 2-vertex connected directed graph.

A k-path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing k-path queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes k-path queries in output-sensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.

Abstract — The problem Min-Power k-Connectivity seeks a power assignment to the nodes in a given wireless ad hoc network such that the produced network topology is k-connected and the total power is the lowest. In this paper, we present several approximation algorithms for this problem. Specifically, we propose a 3k-approximation algorithm for any k ≥ 3, a(k +12H (k))-approximation algorithm for k (2k − 1) ≤ n where n is the network size, a (k +2⌈(k +1)/2⌉)-approximation algorithm for 2 ≤ k ≤ 7, a6-approximation algorithm for k =3, and a 9-approximation algorithm for k =4. Index Terms — k-connectivity, power assignment, wireless ad hoc sensor networks I.

We describe an architecture for optical local area network (LAN) or metropolitan area network (MAN) access. The architecture allows for bandwidth sharing within a wavelength and is robust to both link and node failures. The architecture can be utilized with an arbitrary, link-redundant mesh network (node-redundancy is necessary only to handle all node failures), and assumes neither the use of a star topology nor the ability to embed such a topology within the physical mesh. Reservation of bandwidth is performed in a centralized fashion at a (replicated) head end node, simplifying the implementation of complex sharing policies relative to implementation on a distributed set of routers. Unlike a router, however, the head end does not take any action on individual packets and, in particular, does not buffer packets. The architecture thus avoids the difficulties of processing packets in the optical domain while allowing for packetized shared access of wavelengths. In this paper, we describe the route construction scheme and prove its ability to recover from single link and single node failures, outline a flexible medium access protocol and discuss the implications for implementing specific policies, and propose a simple implementation of the recovery protocol in terms of state machines for per-link devices.

. For any fixed rational parameter t 1, a (tree) t--spanner of a graph G is a spanning subgraph (tree) T in G such that the distance between every pair of vertices in T is at most t times their distance in G. General t--spanners and their variants have multiple applications in the field of communication networks, distributed systems, and network design. In this paper, we combine the two concepts of simple structured, sparse t--spanners and faulttolerance by examining independent tree t--spanners. Given a root vertex r, this is a pair of tree t--spanners, such that the two paths from any vertex to r are edge disjoint or internally vertex dijoint, respectively. It is shown that a pair of independent tree t--spanners can be found in linear time for t ! 3, whereas the problem for arbitrary t 4 is NP--complete. As a less restrictive concept, we also treat tree t--root-spanners, where the distance constraint is relaxed. Here, we show that the problem of finding an independent pair of su...

We present a mathematical model for network routing based on generating paths in a consistent direction. Our development is based on an algebraic and geometric framework for defining a directional coordinate system for real vector spaces. Our model, which generalizes graph st-numberings, is based on mapping the nodes of a network to points in multidimensional space and ensures that the paths generated in di#erent directions from the same source are node-disjoint. Such directional embeddings encode the global disjoint path structure with very simple local information. We prove that all 3-connected graphs have 3-directional embeddings in the plane so that each node outside a set of extreme nodes has a neighbor in each of the three directional regions defined in the plane. We conjecture that the result generalizes to k-connected graphs. We also showthat a directed acyclic graph (dag) that is k-connected to a set of sinks has a k-directional embedding in (k - 1)-space with the sink set as the extreme nodes.

A pair of spanning trees rooted at a vertex r are independent if for every vertex v the pair of unique tree paths from v to the root r are disjoint. This paper presents the first analysis of the path lengths involved in independent spanning trees in 2-edge-connected and 2-vertex-connected graphs. We present upper and lower bounds on the stretch factors of pairs of independent spanning trees, where the stretch factor of a spanning tree is defined to be the maximum ratio between the length of paths in the tree to the root to the length of the shortest path in the graph to the root. We prove that if G is a 2-edge-connected graph with the property that every edge lies on a cycle of size at most h than we can construct in linear time a pair of edge-independent spanning trees whose stretch factors are bounded by O(h). In fact, we prove a more general result, namely that the stretch factor of both independent trees can be bounded by a minimax length of ears with respect to a certain class of ...

uch as buffering, adding packets and dropping packets, or merging packet streams, are done with ease. In optics, however, buffering is onerous. Operations such as retrieving a packet from a traffic stream affect the whole stream. The routing approach we propose seeks to make use of the strengths of optics and avoids operations that are cumbersome or expensive in optics. Our goal is to provide robust and reliable approaches to access at optical data rates. Our design rationale is the following: Avoid buffering. Buffering in the network, with the attendant issues of cost and possibility of overflow, is to be avoided. Therefore, we avoid traffic merging. Traffic splitting, on the other hand, does not require buffering. Do not use a switch unless necessary. While traffic to remote locations should be handled by such switches, local traffic need not be handled by these switches. It is desirable to isolate local traffic from the vagaries of congestion associ

F20.22> 6;6 and K 6;7 are 3-choosable (and simplified the proof that K 5;8 is 3-choosable), thereby completing the proof that n(3) = 14. They also proved n(k) k \Delta n(k \Gamma 2) + 2 k , which improves the previously known upper bounds (from m(k)) when k is even. Zolt'an Furedi proved that K 5;13 is not 3-choosable, so the remaining complete bipartite graphs for which 3-choosability has not been settled are K 5;t for 9 t 12 and K 6;t for 8 t 10. [6] also observes that K k;t is k-choosable if and only if t ! k<F

. We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths. Keywords. independent spanning trees, line digraphs, vertex-connectivity, de Bruijn digraphs, Kautz digraphs, interconnection networks, broadcasting. 1 Introduction Unless stated otherwise each digraph of this paper is nite and may have loops but not multiarcs. Let G be a digraph. Then V (G) and A(G) denote the vertex set and the arc set of G, respectively. Let (u; v) 2 A(G). Then we say that u is adjacent to v, and v is adjacent from u. Also, it is said that (u; v) is incident to v and incident from u. Let (v; w) 2 A(G). Then we say that (u; v) is adjacent to (v; w), and (v; w) is adjacent from (u; v). Let ...