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: We prove that a 3-connected cubic graph contains a cycle through any nine points.

Abstract: The thesis is divided into two parts In the first part, we describe and analyze several new VLSI layouts for the shuffle-exchange graph These include:\ 1) an asymptotically optimal, (N /log N)-area layout for the N-node shuffle-exchange graph, and 2) several practical layouts for small shuffle-exchange graphs The new layouts require substantially less area than previously known layouts and can serve as the basis for designing large scale shuffle-exchange chips In the second part of the thesis, we develop general methods for proving lower bounds on the layout area, crossing number, bisection width and maximum edge length of VLSI networks Among other things, we use these methods to find: 1) an N-node planar graph which has layout area (NlogN) and maximum edge length (N /log N), 2) an N-node graph with an O(N )-separator which has layout area (Nlog N) and maximum edge length (N logN/loglogN), and 3) an -node graph with an O(N )-separator (for < ) which has maximum edge length (N ) The area results indicate that some graphs with O(N )-separators (and, in particular, some planar graphs) do not have linear-area layouts, thus disproving a popular conjecture The edge length bounds indicate that the layouts of some networks must have very long wires (possibly as long as the width of the layout)

Abstract: An algorithm for the determination of an Hamiltonian circuit in a 4-connected planar graph is presented. The timing for this algorithm depends on $n^3 $ (where n is the number of edges in the graph); the storage requirement also depends on $n^3 $. This paper completes the result of Garey, Johnson and Tarjan [SIAM J. Comput., 5 (1976), pp. 704–714] which claims that the problem is NP-complete for 3-connected planar graphs. This algorithm is inspired by the proof of Tutte’s theorem which implies the existence of Hamiltonian circuits in 4-connected planar graphs.

Abstract: If a graph can be embedded in a torus in such a way that all noncontractible cycles have length at least 8, then its vertices may be 5-colored. The conclusion remains true when some noncontractible cycles have length less than 8, if the exceptions are all homotopic. Essentially this hypothesis means that small neighborhoods of the graph are planar. No similar conclusion holds for 4-colorability. Introduction. An r-coloring of a graph G is an assignment of the colors 1, 2, . .. , r to the vertices of G so that any two adjacent vertices are assigned different colors. A graph is said to be r-chromatic if it has an r-coloring but no (r 1)-coloring. It is well known that if G is embedded on the torus then G can be 7-colored [7]. Furthermore every 7-chromatic toroidal graph contains K7 as a subgraph [5]. Such a characterization of r-chromatic toroidal graphs is beyond view for 3 S r < 5 and likely to be difficult for r = 6. Some possible characterizations for 6-chromatic toroidal graphs are suggested in the section of open questions at the end of this paper. One partial result in this direction is that there is a unique 6-chromatic toroidal graph which is regular of degree six [2]-a result which, like the theorem we present here, is based on the Four Color Theorem [3, 4]. Many of the attacks upon the Four Color Problem (including the eventually successful one of finding an unavoidable set of reducible configurations) have been based upon the fundamental premise that most obstructions to 4-coloring are "local" in nature. Thus if G is a toroidal graph which locally appears to be planar it might be reasonable to believe that G is 4-colorable. Surprisingly, this is not always true. Steve Fisk has constructed 5-chromatic graphs that can satisfy any notion of "local planarity" on any surface except the sphere [6]. As his result is ingenious yet simple, we include a proof of it following the proof of our main result. In this paper we intend to prove that locally planar graphs on the torus are 5-colorable. The first step is to find a precise definition which corresponds to the vague notion of locally planar. Let G be a graph embedded in a torus T. A cycle in G is said to be minimal if it has no diagonals in G, and noncontractible, or ncif it is not homotopic to a point. Let C0 be an nc-cycle of minimal length, and among Received by the editors April 22, 1980 and, in revised form, April 22, 1981. Presented at the 83rd summer meeting of the AMS, August 25, 1979. 1980 Mathematics Subject Classification. Primary 05C15.

Abstract: This paper deals with a class of grammars which is called tree-graph grammar (TGG) and its attributed version (ATGG, attributed-tree-graph grammar). The features of this class of grammars include its power to generate useful graphs and its fast parser. Principally, the graphs that can be characterized by these grammars are a subset of hierarchical graphs, including planar graphs. Also these TGG's (and ATGG's) can generate graphs describing, for example, textures, flow charts, circuit diagrams, etc. It is proved that the class of context-free graph grammars (CFGG) is a subclass of TGG's. Also it is shown that TGG can generate graphs which can not be produced by CFGG's, for example, planar graphs. Moreover, a parser for ATGG's is proposed. This parser has time complexity of 0(n4) (n is the number of nodes of the graph plus the number of superimposed nodes), and space complexity of 0(n2). For the special case where the starting node is given, the time complexity is reduced to 0(n3). Finally, an application of ATGG's to the recognition of circuit diagrams is presented.

Abstract: We obtain an asymptotic formula for the number of combinatorially distinct convex polyhedra with n edges. 1. Introduction. The number of combinatorial polyhedra has been studied by many authors, perhaps first of all by Euler. A very interesting account up to 1975 has been given by P. J. Frederico (2). By a well-known theorem of Steinitz a convex polyhedron is combinatorially equivalent to a 3-connected planar map (with more than three edges). B. Griinbaum's proof (3) is the most elegant known to these authors. Indeed Mani (5) has shown that for each 3-connected planar map M there is a convex polyhedron whose 1-skeleton is isomorphic to M and such that every automorphism of M is induced by an isometry of the convex polyhedron. The enumeration of planar maps has progressed considerably following the breakthrough of W. T. Tutte in the early sixties. Tutte (6) has given a survey of the methods and results known up to 1973. Until quite recently most of the progress concerned the enumeration of rooted planar maps. A planar map is said to be rooted when one directed edge in it is distinguished as the root and when two sides of the root are distinguished as "left" and "right". In fact Tutte (7) has shown that the number R(n) of rooted 3-connected planar maps with n edges is asymptotic to

Abstract: The purpose of this paper is to present various results on the said topic, some of which were announced in [24] while others extend earlier results [22, 23] of the author In Section 1 we present non-Hamiltonian simple 3-connected planar graph in which the largest face is a 7-gon; Section 2 extends earlier results on non-Hamiltonian simple 3-connected planar graphs in which there are only two types of faces; Section 3 deals with a curious property of a family of some simple 3-connected planar graphs having only quadrangles and hexagons; we finish by two short remarks, one on a theorem of A Kotzig and another on covering graphs by their Hamiltonian circuits

Abstract: The problem of deciding whether a partial binary operation, a "bin", can be embedded in a semigroup is the associativity problem (for general bins). It is known that it is equivalent to the word problem for (semi)groups and thus unsolvable, even for the class of finite bins. This paper establishes a close association between bins and their wordchains and 3-connected 3-regular planar graphs, or, equivalently convex 3-regular polyhedral nets (skeletons). This permits a constructive approach revealing the combinatorial depth of the associativity problem in detail and leads to a naturally enumerable hierarchy of standard wordchain patterns, of universal bins, and of associative laws. Each bin is a superposition of homomorphic images, i.e. "colourings" of edges, of universal bins. One side result is a purely algebraic equivalent of the 4-colour-theorem. The obtained results open further ways for an efficient search by computer for simplest non-associativity contradictions. It is hoped that they lead to solutions of the associativity problem for further subclasses of bins, further insight into the structure of partial binary operations and of polyhedra and will yield precise measures of presentations for associative systems and their classifications.

Abstract: The toroidal thickness t1(G) of a graph G is the minimum value of k such that G is the union of k graphs each of which is embeddable on a torus. We find t1(Gm), where Gm is the graph obtained from the complete graph Km by removing a Hamiltonian cycle, and we show that t1(Kn(3)) = [1/2n] for many values of n. The method of approach involves the construction of sets of triples related to Skolem triples.

Abstract: It is shown that some classes of cyclically 5-edge-connected cubic planar graphs with only one type of face besides pentagons contain non-Hamiltonian members and have shortness coefficients less than unity.

Abstract: There doesn't exists a finite planar map with all edges having the same length, and each vertex on exactly 5 edges.

Abstract: DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich für den persönlichen, nicht kommerziellen Gebrauch bestimmt. Das Copyright bleibt bei den Herausgebern oder sonstigen Rechteinhabern. Als Nutzer sind Sie sind nicht dazu berechtigt, eine Lizenz zu übertragen, zu transferieren oder an Dritte weiter zu geben. Die Nutzung stellt keine Übertragung des Eigentumsrechts an diesem Dokument dar und gilt vorbehaltlich der folgenden Einschränkungen: Sie müssen auf sämtlichen Kopien dieses Dokuments alle Urheberrechtshinweise und sonstigen Hinweise auf gesetzlichen Schutz beibehalten; und Sie dürfen dieses Dokument nicht in irgend einer Weise abändern, noch dürfen Sie dieses Dokument für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, aufführen, vertreiben oder anderweitig nutzen; es sei denn, es liegt Ihnen eine schriftliche Genehmigung von DigiZeitschriften e.V. und vom Herausgeber oder sonstigen Rechteinhaber vor. Mit dem Gebrauch von DigiZeitschriften e.V. und der Verwendung dieses Dokuments erkennen Sie die Nutzungsbedingungen an.

Abstract: Let {pk}k≥2, k≠4 be a sequence of non-negative integers which satisfies 8 + Σk≥3(k — 4)pk = 0. Then there exists an integer p4 such that there exists a 2-connected planar graph with exactly pkk-gons as faces for all k ≥ 2. This paper determines all such p4 when pk = 0 for k ≥ 5 and determines that there is a constant C ≥ 1 such that for some m ≤ p2 + 1/4p3 + C, there exists a 2-connected planar graph with exactly pk faces for each p4 = m + 2w, w a positive integer. When there exists at least one odd k ≥ 3 for which pk ≠ 0, the coefficient 2 of w in the above equation may be replaced by 1. These conclusions do not hold if the coefficients of p2 and p3 are any smaller than 1 and 1/4, respectively.

Abstract: LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF 1,F 2, …,F r. Letf 1,f 2, …,f r denote the lengths of the cycles bounding the facesF 1,F 2, …,F r respectively. LetC 3(G) be the number of cycles of length three inG. We give bounds onC 3(G) in terms ofp,f 1,f 2, …,f r. WhenG is 3-connected these bounds are bounds for the number of triangles in a polyhedron. We also show that all possible values ofC 3(G) between the maximum and minimum value are actually achieved.

Abstract: Two new generalizations of planar graphs, called quasiplanar and pseudoplanar graphs, are introduced and discussed. It is shown that planar graphs are quasiplanar and these in turn are pseudoplanar. Conversely, a pseudoplanar graph that contains with each arc its reverse arc is quasiplanar. A Kuratowski-type characterization of quasiplanar graphs is given; in it the excluded subgraphs differ from Kuratowski's only by the addition of an edge in the bipartite graph.

Abstract: Three planarity algorithms PA-1, PA-2, and PA-3 are presented which are substantial improvements upon those of Demoucronet al. and Rubin, respectively. The algorithms test the planarity of a graphG by attempting to embed it into the plane step by step. If such an embedding can succeed,G is determined as planar and its planar representation is acquired simultaneously. Among others, PA-3 has been implemented in ALGOL on UNIVAC-1100. The program takes about 13.5 seconds to test a graph of 1000 vertices and 2994 edges.

Abstract: Image transformation and measurements are defined either in the plane (R2) or on regular lattices of points. Each of the two versions is a mathematical model, the former being more adapted to questions such as rotations, similarities, convergence, and relationships with physical properties, the latter being more accessible to numerical treatment. The key idea of the paper is the following: the only notion which has a physical meaning with respect to digitization is the pair: “set model-morphological criterion”. This point of view leads to a definition of digitization linking models to criteria. From this standpoint 3 questions have been considered: Transformations digitizable for the most extensive set-class in morphology (i.e. the closed sets F(R2)). A theorem is given according to which the increasing and upper semi-continuous mapping are digitizable for the closed sets. Transformations preserving honotopy. It is shown that if x e F(R2) and its complement Xc are both open as a compact disk B with radius a > 0, then and only then, their intersections by an hexagonal lattice with spacing ≤ a generate graphs preserve the original homotopy. The digitization of some stereological parameters, in the context of the convex ring model, is studied.

Abstract: Graph analysis is applicable in every situation in which "objects" (points) have relations with (lines toward) other objects and in which the structure of these relations is to be studied. Some examples are: relations between persons (family, friends, commercial relations), relations between functions (e.g., functions fulfilled by the same person), and functions between political parties and states, where a common border can be considered as the relation defining the graph. The aim of graph analysis is the detection of patterns in such networks, the interdependence of patterns formed by different relations, the implications of patterns for the behavior of objects, and the impact of the qualities of the objects on the patterns. Graph analysis, therefore, consists of methods that, on given graphs, try to find: sub graphs (subsets of points in the graph) that meet certain requirements of relations (like strong relations), partial graphs (all points, with a subset of lines) in selecting relations, relations between such subgraphs, detection of trees in the graph (e.g., the shortest spanning tree), and detection of paths (e.g., the shortest path). Of course, the analysis becomes more complicated when: a directed graph is used (lines run in only one direction), the graph is "signed" (each directed line has a positive or negative sign), the points and/or lines are valued (e.g., the "distance" between two points as a value for the connecting line), or more than one type of graph is used at the same time (a multigraph). Also, more than one relation between points may be considered in the same analysis.

Abstract: In a previous paper we have announced that a graph is non-planar if and only if it contains a maximal, strict, compact, odd ring. Little has conjectured that the compactness condition may be removed. Chernyak has now published a proof of this conjecture. However, it is difficult to test a ring for maximality. In this paper we show that for odd rings of size five or greater, the condition of maximality may be replaced by a new one called regularity. Regularity is an easier condition to diagnose than is maximality.

Abstract: Given a formulation of a problem, a compact representation is required both for theoretical purposes -- measuring the complexity of algorithms, and for practical purposes -- data compression. The adjacency lists method for representing graphs is compared to the information theoretic lower bounds, and it is shown to be optimal in many instances. For n-vertex labeled planar graphs the adjacency lists method requires 3nlogn + O(n) bits, a linear algorithm is presented to obtain a 3/2nlogn + O(n) representation while nlogn + O(n) is shown to be the minimum.

Abstract: The virial coefficients Bn of the pressure of a thermodynamic system can be represented in terms of graphs The recently defined overlap graphs are studied in detail Furthermore, the overlap graph representation of the sixth and seventh virial coefficientsB 6 andB 7) is determined