We consider online routing strategies for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing strategies, one that works for all Delaunay triangulations and the other that works for all regular triangulations, (2) a randomized memoryless strategy that works for all triangulations, (3) an O(1) memory strategy that works for all convex subdivisions, (4) an O(1) memory strategy that approximates the shortest path in Delaunay triangulations, and (5) theoretical and experimental results on the competitiveness of these strategies.

Let G be an n-vertex graph with nonnegative weights whose sum is 1 assigned to its vertices, and with no minor isomorphic to a given h-vertex graph H. We prove that there is a set X of no more than h 3/2 n 1/2 vertices of G whose deletion creates a graph in which the total weight of every connected component is at most 1/2. This extends significantly a well-known theorem of Lipton and Tarjan for planar graphs. We exhibit an algorithm which finds, given an n-vertex graph G with weights as above and an h-vertex graph H, either such a set X or a minor of G isomorphic to H. The algorithm runs in time O(h 1/2 n 1/2 m), where m is the number of edges of G plus the number of its vertices. Our results supply extensions of the many known applications of the Lipton-Tarjan separator theorem from the class of planar graphs (or that of graphs with bounded genus) to any class of graphs with an excluded minor. For example, it follows that for any fixed graph H, given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1 / √ log n in polynomial time, find that size exactly and find the chromatic number of G in time 2 O( √ n) and solve any sparse system of n linear equations in n unknowns whose sparsity structure 0 corresponds to G in time O(n 3/2). We also describe a combinatorial application of our result which relates the tree-width of a graph to the maximum size of a Kh-minor in it.

Autonomous robots must be able to learn and maintain models of their environments. Research on mobile robot navigation has produced two major paradigms for mapping indoor environments: grid-based and topological. While grid-based methods produce accurate metric maps, their complexity often prohibits efficient planning and problem solving in large-scale indoor environments. Topological maps, on the other hand, can be used much more efficiently, yet accurate and consistent topological maps are often difficult to learn and maintain in large-scale environments, particularly if momentary sensor data is highly ambiguous. This paper describes an approach that integrates both paradigms: grid-based and topological. Grid-based maps are learned using artificial neural networks and naive Bayesian integration. Topological maps are generated on top of the grid-based maps, by partitioning the latter into coherent regions. By combining both paradigms, the approach presented here gains advantages from both worlds: accuracy/consistency and efficiency. The paper gives results for autonomous exploration, mapping and operation of a mobile robot in populated multi-room environments.

Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and single-source digraphs. We also sketch the proof of NP-completeness of upward planarity testing.

This article describes techniques that have been developed for spatial learning in dynamic environments and a mobile robot system, ELDEN, that integrates these techniques for exploration and navigation in dynamic environments. In this research, we introduce the concept of adaptive place networks, incrementally-constructed spatial representations that incorporate variable-confidence links to model uncertainty about topological adjacency. These networks guide the robot's navigation while constantly adapting to any topological changes that are encountered. ELDEN integrates these networks with a reactive controller that is robust to transient changes in the environment and a relocalization system that uses evidence grids to recalibrate dead reckoning. Footnotes 1 Manuscript received . . . 2 Department of Computer Engineering and Science, Case Western Reserve University, Cleveland, OH, 44106, USA (email: yamauchi@alpha.ces.cwru.edu, URL: http://yuggoth.ces.cwru.edu/yamauchi/index.ht...

We give a strengthening of the closure concept for claw-free graphs introduced by the second author in 1997. The new closure of a claw-free graph G defined here is uniquely determined and preserves the value of the circumference of G. We present an infinite family of graphs with n vertices and 3 2 n \Gamma 1 edges for which the new closure is the complete graph K n . Keywords: closure, cycle closure, claw-free graph, circumference, hamiltonian graph AMS Subject Classifications (1991): 05C45, 05C35 1 1 Introduction We consider finite simple undirected graphs G = (V (G); E(G)). For concepts and notation not defined here we refer the reader to [1]. We denote by c(G) the circumference of G, i.e. the length of a longest cycle in G, by NG (x) the neighborhood of a vertex x in G (i.e., NG (x) = fy 2 V (G)j xy 2 E(G)g), and we denote NG [x] = NG (x) [ fxg. For a nonempty set A ` V (G), the induced subgraph on A is denoted by hAiG , the notation G \Gamma A stands for hV (G) n AiG (if A ...

A tree t-spanner T of a graph G is a spanning tree in which the distance between every pair of vertices is at most t times their distance in G. This notion is motivated by applications in communication networks, distributed systems, and network design. This paper studies graph theoretic, algorithmic and complexity issues about tree spanners. It is shown that a tree 1-spanner, if it exists, in a weighted graph with m edges and n vertices is a minimum spanning tree and can be found in O(m log β(m, n)) time, where β(m, n) = min{i | log (i) n ≤ m/n}. On the other hand, for any fixed t> 1, the problem of determining the existence of a tree t-spanner in a weighted graph is proven to be NP-complete. For unweighted graphs, it is shown that constructing a tree 2-spanner takes linear time, whereas determining the existence of a tree t-spanner is NP-complete for any fixed t ≥ 4. A theorem which captures the structure of tree 2-spanners is presented for unweighted graphs. For digraphs, an O((m+n)α(m, n)) algorithm is provided for

Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization, and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of straight-line representation has not been solved completely. In this paper, we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.

A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

Abstract. Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. Weprovethatif Hhas tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP-complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P � = NP, then no oriented triad H with an NP-complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP-complete Hcolouring problems given in the companion paper has tree duality. 1.