Abstract: Information is not transferred instantaneously; there is always a propagation delay before an output is available as an input to the next computational step. Propagation delay is a function of wire length, so we study the length of edges in planar graphs. We prove matching (to within a constant factor) upper and lower bounds on minimax edge length for four planar embedding problems for complete binary trees. (The results are summarized in Table 1.) Because trees are often subcircuits of larger circuits, these results imply general performance limits due to propagation delay. The results give important information for the popular technique of pipelining.

Abstract: Given a planar graph, we wish to draw it in the plane so that no edges cross. We might be given a particular planarisation (a specification giving the faces of the desired drawing) instead of merely the graph, but this is not required. Given a planarisation and a choice of outermost face, all drawings are in a sense equivalent; indeed, when the drawing is performed on the surface of a sphere instead of on the plane, even the choice of outermost face is irrelevant. To make the problem meaningful, we must introduce further constraints. Two general forms of constraints are: (1) absolute restrictions on the details of the drawing, that is, disallowing or requiring certain features, and (2) weighted restrictions, that is, additional objectives to be met to whatever extent is possible. We first examine the general problem, and look at various constraints that have been found to yield useful drawings in some applications. We then examine in more detail one particular set of constraints, developing a fast algorithm for producing drawings meeting those constraints, and proving some theorems relating to the overall complexity of the problem. Finally, we look at what results are known regarding other variations on the general problem.

Abstract: All known cycle vector space algorithms for listing cycles of a graph are inefficient, and in the worst case they compute all vectors of the cycle space This is a very significant drawback of the cycle space approach In this paper, a cycle vector space algorithm for enumerating all cycles of a planar graph, which produces only cycles of a graph and requires $O(n)$ space and $O(n + nc)$ time (where n and c denote the number of vertices and cycles of a graph, resp), is presented Thus we show that in the class of planar graphs, the cycle space approach can be as efficient as the backtrack algorithms

Abstract: Let N be a planar undirected network with distinguished vertices s, t, a total of n vertices, and each edge labeled with a positive real (the edge's cost) from a set L. This paper presents an algorithm for computing a minimum (cost) s-t cut of N. For general L, this algorithm runs in time O(n log2(n)) time on a (uniform cost criteria) RAM. For the case L contains only integers ≤n0(1), the algorithm runs in time O(n log(n)loglog(n)). Our algorithm also constructs a minimum s-t cut of a planar graph (i.e., for the case L= {1}) in time O(n log(n)).

Abstract: Sufficient conditions on the degrees of a graph are given in order that its line graph have a hamiltonian cycle.

Abstract: Planar embedding with minimal area of graphs on an integer grid is one of the major issues in VLSI Valiant [1981] gave an algorithm to construct a planar embedding for trees in linear area; he also proved that there are planar graphs that require quadratic area We give an algorithm to embed outerplanar graphs in linear area We extend this algorithm to work for every planar graph that has the following property: for every vertex there exists a path of length less than K to the exterior face, where K is a constant Finally, finding a minimal embedding area is shown to be NP-complete for forests, and hence for more general types of graphs

Abstract: Relaxation is applied to the segmentation of closed boundary curves of shapes. The ambiguous segmentation of the boundary is represented by a directed graph structure whose nodes represent segments, where two nodes are joined by an arc if the segments are consecutive along the boundary. A probability vector is associated with each node; each component of this vector provides an estimate of the probability that the corresponding segment is a particular part of the object. Relaxation is used to eliminate impossible sequences of parts, or reduce the probabilities of unlikely ones. In experiments involving airplane shapes, this almost always results in a drastic simplification of the graph with only good interpretations surviving. The approach is also extended to include curve linking and gap filling. A chain coded input image is broken into segments based on a measure of local curvature. Gap completions linking pairs of segments are then proposed and represented in a graph structure. A second graph, whose nodes consist of paths in the above graph, is constructed, and the nodes of the second graph are probabilistically classified as various object parts. Relaxation is then applied to increase the probability of mutually supporting classifications, and decrease the probability of unsupported decisions. A modified relaxation process using information about the size, spatial position, and orientation of the object parts yielded a high degree of disambiguation.

Abstract: Adominating cycle of a graph lies at a distance of at most one from all the vertices of the graph. The problem of finding the minimum size of such a cycle is proved to be difficult even when restricted to planar graphs. An efficient algorithm solving this problem is given for the class of two-connectedouterplanar graphs, in which all vertices lie on the exterior face in a plane embedding of the graph.

Abstract: The fact that several inlportant combinatorial optimiz.ation problems are NP-colnplete has motivated research on the worst-case analysis of approximation heuristics for these problems [Jol, GIl, Ch). TIlcse inve.stigations have produced some very interesting results" and considerable insight has been gained by no\v into the power and limitations of existing techniques. The most inlportnnt paradigm in this area is bin packing [JDGGU, J02, Yao, GJ2, GJ3] and its generalizations [GOJY, CGJT). '[his is so because of the elegance and depth of the cOlnbinatorial arguments employed in the proofs of the upper bounds, and the intricate constnlctions of exanlples that achieve them. Despite the presence of the unifying cotlcept of a weighting [unction, the arguments are usually ingenious yet ad hoc, and the construction of worst-case exmnp)es is largely decoupled from the upper bounding process. In this paper we present a fanlily of results concerning certain extrenlal properties of planar graphs. In particular we show the follo\ving: (1) The greedy heuristic (i.e., repeatedly pick the node with smallest degree and delete its neighborhood) applied to a planar graph with n nodes yields an independent set of size at least 4n/21. (2) l'he greedy heuristic yields an independent set at )etlst 23/63 times the optimum. (3) A planar graph with n nodes and minimum degree 3 has ahvays a nzatching with fewer than n/3 free nodes.

Abstract: The connection graph proof procedure [1, 2] allows us to remove a clause from the graph, if it is a tautology or a pure clause, i.e. if it is a clause which contains a literal which is not connected to any other literal in the graph.

Abstract: The problem of determining which 4-regular graphs are decomposable into two edge-disjoint Hamilton cycles was first considered by Kotzig [4], who proved that a 3-regular graph is hamiltonian if and only if its edge graph (that is, line graph) has a hamiltonian decomposition. With the aid of this theorem, it is an easy matter to construct examples of 4-regular graphs which fail to have such a decomposition. Later, and presumably unaware of Kotzig's work (which was published in Slovak), Nash-Williams [7] conjectured that every 4-connected 4-regular graph admits a hamiltonian decomposition. Meredith [6] disproved this conjecture by constructing a nonhamiltonian 4-connected 4-regular graph. Gr/inbaum and Zaks [3] then proposed a weaker version of Nash-Williams' conjecture, restricted to planar graphs. (Recall that, by a theorem of Tutte [8], all 4-connected planar graphs are hamiltonian.) Counterexamples to this weaker conjecture were found by Gr/inbaum and Malkevitch [2] and also by Martin [5], who independently rediscovered Kotzig's theorem. The purpose of this note is to point out that Grinberg's necessary condition for a plane graph to be hamiltonian [1] can be used to derive a similar necessary condition for a 4-regular plane graph to admit a hamiltonian decomposition. To simplify the statement of Grinberg's theorem and its subsequent application, we make the following definition. If G is any plane graph with face set F, let g : 2 ~ ~ N be defined by

Abstract: Algorithms in computational geometry are of increasing importance in computer-aided design, for example, in the layout of integrated circuits. The efficient computation of the intersection of several superimposed figures is a basic problem. Plane figures defined by points connected by straight line segments are considered, for example, polygons (not necessarily simple) and maps (embedded planar graphs). The regions into which the plane is partitioned by these intersecting figures are to be processed in various ways such as listing the boundary of each region in cyclic order or sweeping the interior of each region. Let m be the total number of points of all the figures involved and s be the total number of intersections of all line segments. A two plane-sweep algorithm that solves the problems above is presented; in the general case (non convexity) in time O((n+s)log-n) and space O(n+s); when the regions of each given figure are convex, the same can be achieved in time O(n log n +s) and space O(n)

Abstract: : The adjacency map is a data structure (a tree) used to solve the following problem: given a set of parallel segments in the plane and a point p, find the segments closest to p among those intersected by the straight line through p, perpendicular to the common direction of the segments. The search is performed in the repetitive mode, so that preprocessing is convenient. The problem considered is a particular case of planar point location for which algorithms are known (Lipton-Tarjan, Kirkpatrick), which make use of data structures constructed in time 0(nlogn), searched in time 0(logn), and stored in space 0(n). Though asymptotically optimal, the previous algorithms are not very practical. More practical algorithms have been proposed (Preparata, Preparata-Lipski), which use 0(nlogn) space. In this thesis a modification of these algorithm is presented for the adjacency map, and the worst case analysis is performed. The technique is easily extensible to general planar graphs. It is conjectured that, under reasonable assumptions on the input distribution, the new algorithm takes expected linear storage. (Author)

Abstract: Preface H. N. V. Temperley 1. On the abstract group of automorphisms L. Babai 2. A tour through tournaments or bipartite and ordinary tournaments: a comparative survey Lowell W. Beineke 3. Shift register sequence Henry Beker and Fred Piper 4. Random graphs Bela Bollobas 5. Recent results in graph decompositions F. R. K. Chung and R. L. Graham 6. The geometry of planar graphs Branko Grunbaum and G. C. Shephard 7. Some connections between designs and codes F. J MacWilliams 8. Counting graphs with a duality property R. W. Robinson 9. Ovals in a projective plane of order 10. John G. Thompson

Abstract: The object of this thesis is to investigate the Reconstruction Problem for planar graphs. This study naturally leads to related topics concerning certain nonplanar graphs and the use of their embeddings on appropriate surfaces to reconstruct them. The principal aim of this work is to find new techniques of reconstruction and to increase the number of classes of graphs known to be reconstructible. In achieving this aim, various important properties of graphs, such as connectivity and uniqueness of embeddings, are explored, and new results on these topics are obtained. Part I, which consists of three chapters, contains a historic a l, non-technical introduction and general graph-theoretical definitions, notation and results. Some new concepts in reconstruction are also presented, notably the idea of reconstructor sets. Part II of the thesis deals with the vertex-reconstruction of maximal planar graphs: Chapter 4 is concerned with the vertex-recognition of maximal planarity, whereas Chapter 5 deals with the vertex-reconstruction. Part III deals with edge-reconstruction: planar graphs with minimum valency 5 and 4-connected planar graphs are reconstructed in Chapters 6 and 7 respectively. In Chapter 7, extensive use is made of the concept of reconstructor sets introduced in Chapter 3. This chapter also contains a brief discussion on the reconstruction of graphs from edge-contracted subgraphs, a problem which, in certain cases, can be regarded as dual to the Edge-reconstruction Problem. Part IV is concerned with extending the results and techniques of the previous chapters to nonplanar graphs. Chapter 8 discusses where the previous techniques fa il, and indicates where new methods are needed. In Chapter 9, all graphs which triangulate some surface and have connectivity 3 are edge-reconstructed. Certain graphs which triangulate the torus or the projective plane are also shown to be weakly vertex-reconstructible. Chapter 10 deals with the edge-reconstruction of all graphs which triangulate the projective plane. The Appendix proves a conjecture of Harary on the cutvertex-reconstruction of trees. One technique used here ties up with a method employed in previous chapters on edge-reconstruction.

Abstract: A Directional Nearest-Neighbor graph is defined on a finite set of points in the plane by drawing an arc from each point X to its nearest neighbor in each of r divisions of the plane relative to X. We prove that Directional Nearest-Neighbor graphs having $r = 4$ are strongly connected.

Abstract: It has been communicated by P. Manca in this journal that all 4-regular connected planar graphs can be generated from the graph of the octahedron using simple planar graph operations. We point out an error in the generating procedure and correct it by including an additional operation.

Abstract: Given a planar straight line graph G with n vertices and a point $P_0 $, locating $P_0 $ means to find the region of the planar subdivision induced by G which contains $P_0 $. Recently, Lipton and Tarjan presented a brilliant but extremely complex point location algorithm which runs in time $O(\log n)$ on a data structure using $O(n)$ storage. This paper presents a practical algorithm which runs in less than $6\lceil {\log _2 n} \rceil $ comparisons on a data structure which uses $O(n\log n)$ storage, in the worst case. The method rests crucially on a simple partition of each edge of G into $O(\log n)$ segments.