Showing papers on "Planarity testing published in 1978"

Algorithms for Loop Matchings

Abstract: A simplified (two-base) version of the problem of planar folding of long chains (e.g., RNA and DNA biomolecules) is formulated as a matching problem. The chain is prescribed as a loop or circular sequence of letters A and B, n units long. A matching here means a set of A-B base pairings or matches obeying a planarity condition: no two matches may cross each other if drawn on the interior of the loop. Also, no two adjacent letters may be matched. We present a dynamic programming algorithm requiring $O( {n^3 } )$ steps and $O( {n^2 } )$ storage which computes the size of the maximum for the given A-B base sequence and which also allows reconstructing a particular folded form of the original string which realizes the maximum matching size. The algorithm can be adapted to deal with sequences with larger alphabets and with weighted matchings.An algorithm is also presented for a modified problem closer to the biochemical problem of interest: We demand that every match must be adjacent to another match, forcing ...

Graph theoretic heuristics for the plant layout problem

Abstract: The plant layout problem is an important industrial problem that remains unsolved. Seppanen and Moore considered an important subproblem: that of the optimal specification of which facilities are to be adjacent in the final layout without regard to the area or shape of the individual facilities. They present a string processing heuristic which relies on using a test for graph planarity. This paper further develops their graph theoretic formulation of the problem. This development is used to present two procedures which usually yield near-optimal solutions. The procedures circumvent the considerable difficulty of graph planarity testing by working entirely within a family of graphs known to be planar and also to contain the solution. Ways of improving suboptimal solutions are discussed. Computational experience in using the methods on a variety of problems suggests that the methods are efficient.

On the Principal Edge Tripartition of a Graph

Abstract: Publisher Summary This chapter presents a general study of a natural partition of the edges of a graph G into three classes—the principal tripartition of G, which is defined canonically from the cycle space of G . For ease of visualization, the graph-theoretic concepts of cycles and cocycles are employed, but unless otherwise stated, the results hold also for binary matroids. The chapter presents proof of a number of theorems concerning this principal tripartition and the results applying them to obtain a criterion for the planarity of a graph to solve a conjecture of Gauss concerning the sequence of crossing points of a closed curve. The tripartition can also be used to obtain many “Parity Theorems”—theorems concerning the parity of certain numbers associated with a graph, such as the number of spanning trees. A Parity theory for graphs may be regarded as promising for networks engineering.

On the Characterization of Balancing Processes of Social Systems and the Derivation of the Minimal Balancing Processes

Abstract: The psychological notion of cognitive balance by Cartwright and Harary offers a well-organized mode for consideration of systems theoretic approaches to social problems. The balancing processes of a social system, which transform an unbalanced system to a balanced one by changing the cognitive relations, are characterized in terms of signed graphs. First, the characterization of the balancing processes of the whole system is reduced to the characterization of each subsystem, which is obtained by block decomposition of the whole system. Second, the characterization is made in two different ways according to the planarity of the graph structure of the systems. For systems with planar graph structure, the characterization is effectively made by introducing the notion of dual graph. For systems with general graph structure, the balancing processes are characterized in terms of adjacency matrices. In each case, an efficient algorithm which derives the minimal balancing processes is presented. Finally, a quantitative approach incorporating the relative importance of each relation is examined.

Spectra and structure of small ring compounds. XXXIX.a) Barrier to interconversion in selenacyclopentane

Abstract: The Raman spectrum of selenacyclopentane in the gaseous phase was observed from 25 to 4000 cm−1. A series of Q branches at 231, 238, and 244 cm−1 has been assigned to transitions of the radial ring mode of an extremely hindered pseudorotator with a planar intermediate. Based on their relative intensities, these bands have been assigned as the 1←0, 2←1, and 3←2 radial transitions, respectively. For the two out‐of‐plane ring modes, we have calculated a potential surface of the form V (q,φ) =Aq4 +Bq2+Cq2 cos2φ+Dq4 cos2φ+Eq4 cos4φ. The barrier to planarity for the twisted conformer is found to be of the same order as in one‐dimensional treatment.

On the planarity of hypergraphs

Abstract: Two definitions for the planarity of a hypergraph will be given. These definitions are related to the problem of modelling circurs for circuit layout. It is shown that if a hypergraph is planar according to the first definition, then it is planar according to the second definition, but the reverse does not hold. Testing the planarity can be done in linear time by using the first definition, but requires an enumeration using the second definition. The more accurate model for the circuit layout problem uses the second definition.

Cellular Graph Acceptors, 4.

Abstract: : Diameter-time algorithms are presented for recognition of rectangular and square arrays, Eulerian graphs, bipartite and complete bipartite graphs, stars, and wheels by cellular d-graph acceptors. Slower algorithms are given for construction of a depth-first spanning tree (area time) and for identification of cut nodes, borders, and central points (diameter'area time). The recognition of planarity is also discussed. (Author)

Vibrational spectra, structure, and barrier to planarity of perfluoro-1,3-butadiene

Abstract: The infrared spectra of gaseous and solid perfluoro‐1,3‐butadiene have been recorded from 4000 to 30 cm−1. Raman spectra of this compound have also been obtained for the gaseous, liquid, and solid phases. The vibrational data are shown to be consistent with a gauche (C2) structure for this molecule and a vibrational assignment has been proposed. Analysis of the torsional frequencies observed in both the Raman and far infrared spectra yields a barrier to planarity at the trans conformation of 986±150 cm−1 (2.85 kcal/mole).