Abstract: INTRODUCTION A Systematic Design Methodology Links and Joints Kinematic Chains, Mechanisms, and Machines Kinematics of Mechanisms Planar, Spherical, and Spatial Mechanisms Kinematic Inversions BASIC CONCEPT OF GRAPH THEORY Definitions Tree Planar Graph Spanning Trees and Fundamental Circuits Euler's Equation Topological Characteristics of Planar Graphs Matrix Representation of Graphs Contracted Graphs Dual Graphs STRUCTURAL REPRESENTATIONS OF MECHANISMS Functional Schematic Representation Structural Representation Graph Representation Matrix Representation STRUCTURAL ANALYSIS OF MECHANISMS Correspondence between Mechanisms and Graphs Degrees of Freedom Loop Mobility Criterion Lower and Upper Bounds on the Number of Joints on a Link Link Assortments Partition of Binary Link Chains Structural Isomorphism Permutation Group and Group of Automorphisms Identification of Structural Isomorphism Partially Locked Kinematic Chains ENUMERATION OF GRAPHS OF KINEMATIC CHAINS Enumeration of Contracted Graphs Enumeration of Conventional Graphs Atlas of Graphs of Kinematic Chains CLASSIFICATION OF MECHANISMS Planar Mechanisms Spherical Mechanisms Spatial Mechanisms EPICYCLIC GEAR TRAINS Structural Characteristics Buchsbaum-Freudenstein Method Genetic Graph Approach Parent Bar Linkage Method Mechanism Pseudo Isomorphisms Atlas of Epicyclic Gear Trains Kinematics of Epicyclic Gear Trains AUTOMOTIVE MECHANISMS Variable-Stroke Engine Mechanisms Constant-Velocity Shaft Couplings Automatic Transmission Mechanisms Canonical Graph Representation of EGMs Atlas of Epicyclic Gear Transmission Mechanisms ROBOTIC MECHANISMS Parallel Manipulators Robotic Wrist Mechanisms APPENDICES: A. Solving m Equations in n unknowns B. Atlas of Contracted Graphs C. Atlas of Graphs of Kinematic Chains D. Atlas of Planar Bar Linkages E. Atlas of Spatial One-dof Kinematic Chains F. Atlas of Epicyclic Gear Trains G. Atlas of Epicyclic Gear Transmission Mechanisms NOTE: Introduction at the beginning of Chapters 1,3-9 Summary at the end of Chapters 1-6,8-9

Abstract: Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulations.

Abstract: The purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi-isometric to the hyperbolic plane.

Abstract: A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n/2. The bisection cost is the number of edges connecting the two sets. Finding the bisection of minimum cost is NP-hard. We present an algorithm that finds a bisection whose cost is within ratio of O(log/sup 2/ n) from the optimal. For graphs excluding any fixed graph as a minor (e.g. planar graphs) we obtain an improved approximation ratio of O(log n). The previously known approximation ratio for bisection was roughly /spl radic/n.

Abstract: The standard serial algorithm for strongly connected components is based on depth first search, which is difficult to parallelize. We describe a divide-and-conquer algorithm for this problem which has significantly greater potential for parallelization. For a graph with n vertices in which degrees are bounded by a constant, we sho w the expected serial running time of our algorithm to be O(n log n).

Abstract: V.G. Vizing conjectured in 1968 that every planar graph with maximum degree 6 or 7 is of class 1. This paper shows that, for planar graphs with maximum degree 7, Vizing's conjecture is true.

Abstract: 1 Introduction.- 1.1 Graphs, Digraphs and Networks.- 1.2 Classifying Problems.- 1.3 Seeking Solutions.- 2 Graphs.- 2.1 Graphs and Subgraphs.- 2.2 Vertex Degrees.- 2.3 Paths and Cycles.- 2.4 Regular and Bipartite Graphs.- 2.5 Case Studies.- Four Cubes Problem.- Social Networks.- Exercises 2.- 3 Eulerian and Hamiltonian Graphs.- 3.1 Exploring and Travelling.- 3.2 Eulerian Graphs.- 3.3 Hamiltonian Graphs.- 3.4 Case Studies.- Dominoes.- Diagram-Tracing Puzzles.- Knight's Tour Problem.- Gray Codes.- Exercises 3.- 4 Digraphs.- 4.1 Digraphs and Subdigraphs.- 4.2 Vertex Degrees.- 4.3 Paths and Cycles.- 4.4 Eulerian and Hamiltonian Digraphs.- 4.5 Case Studies.- Ecology.- Social Networks.- Rotating Drum Problem.- Ranking in Tournaments.- Exercises 4.- 5 Matrix Representations.- 5.1 Adjacency Matrices.- 5.2 Walks in Graphs and Digraphs.- 5.3 Incidence Matrices.- 5.4 Case Studies.- Interval Graphs.- Markov Chains.- Exercises 5.- 6 Tree Structures.- 6.1 Mathematical Properties of Trees.- 6.2 Spanning Trees.- 6.3 Rooted Trees.- 6.4 Case Study.- Braced Rectangular Frameworks.- Exercises 6.- 7 Counting Trees.- 7.1 Counting Labelled Trees.- 7.2 Counting Binary Trees.- 7.3 Counting Chemical Trees.- Exercises 7.- 8 Greedy Algorithms.- 8.1 Minimum Connector Problem.- 8.2 Travelling Salesman Problem.- Exercises 8.- 9 Path Algorithms.- 9.1 Fleury's Algorithm.- 9.2 Shortest Path Algorithm.- 9.3 Case Study.- Chinese Postman Problem.- Exercises 9.- 10 Paths and Connectivity.- 10.1 Connected Graphs and Digraphs.- 10.2 Menger's Theorem for Graphs.- 10.3 Some Analogues of Menger's Theorem.- 10.4 Case Study.- Reliable Telecommunication Networks.- Exercises 10.- 11 Planarity.- 11.1 Planar Graphs.- 11.2 Euler's Formula.- 11.3 Cycle Method for Planarity Testing.- 11.4 Kuratowski's Theorem.- 11.5 Duality.- 11.6 Convex Polyhedra.- Exercises 11.- 12 Vertex Colourings and Decompositions.- 12.1 Vertex Colourings.- 12.2 Algorithm for Vertex Colouring.- 12.3 Vertex Decompositions.- Exercises 12.- 13 Edge Colourings and Decompositions.- 13.1 Edge Colourings.- 13.2 Algorithm for Edge Colouring.- 13.3 Edge Decompositions.- Exercises 13.- 14 Conclusion.- 14.1 Classification of Problems.- 14.2 Efficiency of Algorithms.- 14.3 Another Classification of Problems.- Suggestions for Further Reading.- Appendix: Methods of Proof.- Computing Notes.- Solutions to Computer Activities.- Solutions to Problems in the Text.

Abstract: We give nontrivial bounds for the inductiveness or degeneracy of power graphs Gk of a planar graph G. This implies bounds for the chromatic number as well, since the inductiveness naturally relates to a greedy algorithm for vertex-coloring the given graph. The inductiveness moreover yields bounds for the choosability of the graph. We show that the inductiveness of a square of a planar graph G is at most $\lceil 9\Delta /5 \rceil$, for the maximum degree $\Delta$ sufficiently large, and that it is sharp. In general, we show for a fixed integer $k\geq1$ the inductiveness, the chromatic number, and the choosability of Gk to be $O(\Delta^{\lfloor k/2 \rfloor})$, which is tight.

Abstract: We present an algorithm for computing the domination number of a planar graph that uses O(c√kn) time, where k is the domination number of the given planar input graph and c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number k is O(√k), and that such a tree decomposition can be found in O(√kn) time. The same technique can be used to show that the DISK DIMENSION problem (find a minimum set of faces that cover all vertices of a given plane graph) can be solved in O(c1√k n) time for c1 = 26√34. Similar results can be obtained for some variants of DOMINATING SET, e.g., INDEPENDENT DOMINATING SET.

Abstract: The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any $n$-vertex, connected graph is at least $\bigl(1+o(1)\bigr)n\log n$ and at most $\bigl(1+o(1)\bigr)\frac{4}{27}n^3$. This paper proves that for bounded-degree planar graphs the cover time is at least $c n(\log n)^2$, and at most $6n^2$, where $c$ is a positive constant depending only on the maximal degree of the graph. The lower bound is established via use of circle packings.

Abstract: We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp, and we present all nonhamiltonian examples of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vertices. The fact that all 3-connected cubic planar graphs on at most 176 vertices and with face size at most 6 are hamiltonian is also verified.

Abstract: This work is motivated bythe problem of placing pressure-meters in fluid networks. The problem is formallydefined in graph-theoretic terms as follows. Given a graph, find a cotree (complement of a tree) incident upon the minimum number of vertices. We show that this problem is NP-hard and MAX SNP-hard. We design an algorithm with an approximation factor of 2 + � for this problem for anyfixed �> 0. This approximation bound comes from the analysis of a local search heuristic, a common practical optimization technique that does not often allow formal worst- case analy sis. The algorithm is made veryefficient byfinding restrictive definitions of the local neighborhoods to be searched. We also exhibit a polynomial time approximation scheme for this problem when the input is restricted to planar graphs.

Abstract: Many classes of valid and facet-inducing inequalities are known for the family of polytopes associated with theSymmetric Travelling Salesman Problem (STSP), includingsubtour elimination, 2-matching andcomb inequalities. For a given class of inequalities, anexact separation algorithm is a procedure which, given an LP relaxation vectorx*, finds one or more inequalities in the class which are violated byx*, or proves that none exist. Such algorithms are at the core of the highly successfulbranch-and-cut algorithms for the STSP. However, whereas polynomial time exact separation algorithms are known for subtour elimination and 2-matching inequalities, the complexity of comb separation is unknown.Apartial answer to the comb problem is provided in this paper. We define ageneralization of comb inequalities and show that the associated separation problem can be solved efficiently when the subgraph induced by the edges withx*e>0 is planar. The separation algorithm runs in O( n3) time, wheren is the number of vertices in the graph.

Abstract: The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex, connected graph is at least (1+o(1)) n log(n) and at most (1+o(1))(4/27)n^3. This paper proves that for bounded-degree planar graphs the cover time is at least c n(log n)^2, and at most 6n^2, where c is a positive constant depending only on the maximal degree of the graph. The lower bound is established via use of circle packings.

Abstract: The Frequency Assignment Problem (FAP) in radio networks is the problem of assigning frequencies to transmitters exploiting frequency reuse while keeping signal interference to acceptable levels. The FAP is usually modelled by variations of the graph coloring problem. The Radiocoloring (RC) of a graph G(V, E) is an assignment function Φ : V → IN such that |Φ(u) - Φ(v)| ≥ 2, when u, v are neighbors in G, and |Φ(u) - Φ(v)| ≥ 1 when the minimum distance of u, v in G is two. The discrete number and the range of frequencies used are called order and span, respectively. The optimization versions of the Radiocoloring Problem (RCP) are to minimize the span or the order. In this paper we prove that the min span RCP is NP-complete for planar graphs. Next, we provide an O(nΔ) time algorithm (|V| = n) which obtains a radiocoloring of a planar graph G that approximates the minimum order within a ratio which tends to 2 (where Δ the maximum degree of G). Finally, we provide a fully polynomial randomized approximation scheme (fpras) for the number of valid radiocolorings of a planar graph G with λ colors, in the case λ ≥ 4Δ + 50.

Abstract: One of the major goals of computer vision is the research and the development of flexible methods for shape description. A large group of shape description techniques is given by heuristic approaches, which yield acceptable results in the description of simple shapes and regions. In this case, objects are represented by a planar graph with nodes symbolizing subregions from region decomposition, and region shape is then described by the graph properties. In the paper, the angular bisector network (ABN), a descriptor of polygonal shape, is used to automatically detect intersections between neurites of cell structures. Some properties of the ABN, such as linear algebraic complexity, easy extraction of characteristic points, etc., are very useful and experimental results are promising.

Abstract: We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(logn) time CRCW PRAM algorithm [22], although planarity checking for degree-three graphs had been shown to be in SL [23, 20].

Abstract: In the graph-searching problem, initially a graph with all the edges contaminated is presented. The objective is to obtain a state of the graph in which all the edges are simultaneously cleared by using the least number of searchers. Two variations of the graph-searching problem are considered. One is edge searching, in which an edge is cleared by moving a searcher along this edge, and the other is node searching, in which an edge is cleared by concurrently having searchers on both of its two endpoints.

Abstract: We present an experimental study in which we compare the state-of-the-art methods for compacting orthogonal graph layouts. Given the shape of a planar orthogonal drawing, the task is to place the vertices and the bends on grid points so that the total area or the total edge length is minimised. We compare four constructive heuristics based on rectangular dissection and on turn-regularity, also in combination with two improvement heuristics based on longest paths and network flows, and an exact method which is able to compute provable optimal drawings of minimum total edge length. We provide a performance evaluation in terms of quality and running time. The test data consists of two test-suites already used in previous experimental research. In order to get hard instances, we randomly generated an additional set of planar graphs.

Abstract: We study the parallel dynamics of a class of Kauffman boolean nets such that each vertex has a binary state machine {AND, OR} as local transition function. We have called this class of nets AON. In a finite, connected and undirected graph, the transient length, attractors and its basins of attraction are completely determined in the case of only OR (AND) functions in the net. For finite, connected and undirected AON, an exact linear bound is given for the transient time using a Lyapunov functional. Also, a necessary and sufficient condition is given for the diffusion problem of spreading a one all over the net, which generalizes the primitivity notion on graphs. This condition also characterizes its architecture. For finite, strongly connected and directed AON a non-polynomial time bound is given for the transient time and for the period on planar graphs, together with an example where this transient time and period are attained. Furthermore, on infinite but finite connected, directed and non planar AON we simulate an universal two-register machine, which allows us to exhibit universal computing capabilities.

Abstract: Akers et al. proposed an interconnection topology, the star graph, as an alternative to the popular n-cube. Cheng et al. proposed the split-star as an alternative to the star graph and a companion graph to the alternating group graph proposed by Jwo et al. Star graphs, alternating group graphs, and split-stars are advantageous over n-cubes in many aspects. Day and Tripathi proposed an assignment of directions to the edges of the star graph and showed that the resulting directed graph is strongly connected and has a simple routing algorithm. In this paper, we give simple routing algorithms for a proposed orientation of alternating group graphs and split-stars. The resulting directed graphs are not only strongly connected but they have maximal arc-fault tolerance and a small diameter. © 2000 John Wiley & Sons, Inc.

Abstract: In 1967 Kasteleyn introduced a powerful method for enumerating the 1-factors of planar graphs. In fact his method can be extended to graphs which permit an orientation under which every alternating circuit is clockwise odd. Graphs with this property are called {\it Pfaffian}. Little characterised Pfaffian bipartite graphs in terms of forbidden subgraphs in 1975. We extend his characterisation to near bipartite graphs.

Abstract: A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type (e-x2), that is, Gaussian. We exhibit here a new class of "universal" phenomena that are of the exponential-cubic type (eix3), corresponding to nonstandard distributions that involve the Airy function. Such Airy phenomena are expected to be found in a number of applications, when confluences of critical points and singularities occur. About a dozen classes of planar maps are treated in this way, leading to the occurrence of a common Airy distribution that describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs.