Showing papers on "Planarity testing published in 1979"

Graph Algorithms

Abstract: Shimon Even's Graph Algorithms, published in 1979, was a seminal introductory book on algorithms read by everyone engaged in the field. This thoroughly revised second edition, with a foreword by Richard M. Karp and notes by Andrew V. Goldberg, continues the exceptional presentation from the first edition and explains algorithms in a formal but simple language with a direct and intuitive presentation. The book begins by covering basic material, including graphs and shortest paths, trees, depth-first-search, and breadth-first search. The main part of the book is devoted to network flows and applications of network flows, and it ends with chapters on planar graphs and testing graph planarity.

Maximum Flow in Planar Networks

Abstract: Efficient algorithms for finding maximum flow in planar networks are presented These algorithms take advantage of the planarity and are superior to the most efficient algorithms to date If the source and the terminal are on the same face, an algorithm of Berge is improved and its time complexity is reduced to $O(n\log n)$ In the general case, for a given $D > 0$ a flow of value D is found if one exists; otherwise, it is indicated that no such flow exists This algorithm requires $O(n^2 \log n)$ time If the network is undirected a minimum cut may be found in $O(n^2 \log n)$ time All algorithms require $O(n)$ space

Some Ramsey-type numbers and the independence ratio

Abstract: If each of k, m, and n is a positive integer, there is a smallest positive integer r = rk(m, n) with the property that each graph G with at least r vertices, and with maximum degree not exceeding k, has either a complete subgraph with m vertices, or an independent subgraph with n vertices. In this paper we determine r3(3, n) = r(n), for all n. As a corollary we obtain the largest possible lower bound for the independence ratio of graphs with maximum degree three containing no triangles. From the work of Brooks [2] it follows that if G is a graph with maximum degree k containing no complete graph on k + 1 vertices, then the independence ratio of G is at least \/k. In case G has no complete graph on k vertices, Albertson, Bollobas, and Tucker [1] proved this ratio is larger than \/k, with only two exceptions. And they conjectured that for k = 3, with the additional assumption of planarity, this ratio is bounded away from 1/3. Fajtlowicz [3] verified their conjecture, even without assuming planarity, showing that each cubic graph without triangles has independence ratio at least 12/35. In addition, he displayed a graph in which the independence ratio is exactly 5/14. It follows from our main theorem that 5/14 is a lower bound for the independence ratio in the case k = 3, and in light of Fajtlowicz' graph, 5/14 is the best possible lower bound. In what follows, all graphs will be finite symmetric graphs with no loops and no multiple edges. If G is a graph, then v(G) and e(G) will be the numbers of vertices and edges of G. If M is a set of vertices of G, no two of which are joined by an edge, then M is called independent. The number of vertices in a largest independent vertex set in G will be denoted i(G). A cycle with « vertices will be denoted C„. Proposition I. If G is a graph in which each vertex has degree two or degree three, and if v(G) is odd, then either there is a vertex of degree two both of whose neighbors are of degree two, or else there is a vertex of degree two both of whose neighbors are of degree three. Received by the editors November 6, 1978. AMS (MOS) subject classifications (1970). Primary 05C99.

Infrared spectra of AlCl2Br, AlClBr2, and AlBr3 in solid argon

Abstract: We report the ir spectra of AlCl2Br, AlClBr2, and AlBr3. It is pointed out that the vibrational spectra of AlCl2Br and AlClBr2 can provide a novel and sensitive test for small deviations from planarity of AlCl3. The observed spectra imply a planar D3h structure for each of these trihalides.

A Linear Time Planarity Algorithm for 2-Complexes

Abstract: A linear time algorithm to decide whether a given finite 2- complex is planar is described. Topological results of Gross, Harary and Rosen are the mathematical basis for the algorithm. Optimal running time is achieved by constructing various lists simultaneously and keeping their orderings compatible. If the complex is simplical with p vertices, then the algorithm has O(p) time and space bounds. The algorithm uses depth-first search both in application of the graph planarity algorithm of Hopcroft and Tarjan and elsewhere

The low frequency Raman spectrum and barrier to planarity of 2‐cyclohexene‐1‐one.

Abstract: The low frequency Raman spectrum of gaseous 2‐cyclohexene‐1‐one has been obtained at the full vapor pressure at ∼110°C using the 488 nm line of an argon ion laser at ∼2W power. 2‐cyclohexene‐1‐one has been found to exist with all of the heavy atoms except C5 essentially in a plane. The series of Q‐branch transitions in the Raman spectrum between 140–250 cm−1 have been assigned as Δv=2 transitions of a one‐dimensional double‐minimum oscillator. The vibarational mode is the motion of the CH2 group at the 5 position through the plane of the heavy atoms. The potential function in a reduced coordinate Z [V (cm−1) =27.49(Z4−11.67 Z2)] yields a barrier to planarity of 935±70 cm−1 (2.67±0.2 kcal/mole, 11.2±0.8 kJ/mole).

INDO calculations on the [Be3O(HCO2)3]+ ion configuration

Abstract: SCF–MO–LCAO–INDO calculations show that the total energy of the [Be3O(HCO2)3]+ ion, which is the base peak in the mass spectra of the beryllium μ4-oxoformato complex, depends on the deviation of the ion configuration from planarity. The planar form is shown to have the lowest energy. The results obtained confirm the suggestion that steric interactions are the major factor governing fragmentations of beryllium μ4-oxocarboxylato complexes under electron impact.

Vanishing graphs, planarity, and reggeization

Abstract: An infinite class of nonplanar skeleton graphs is found to vanish in any non-Abelian gauge theory. Thus, the dominance of planar graphs is enhanced, particularly in processes where some momenta are very large.

On the planarity of the NSi3 skeleton in the trisilylamine molecule. A normal coordinate analysis involving complex symmetry coordinates

Abstract: Normal coordinate calculations have been carried out for the N(SiH3)3 molecule. A model with C3h symmetry was used, which is based on a planar NSi3 skeleton. The presence of complex numbers for the...

Device for attaching planar plate member

Abstract: In fabricating very-high-resolution devices with a lithographic tool such as an EBES machine, it is crucial that the planar mask or wafer members utilized therein be loaded into and maintained in the work chamber of the machine in a way that minimally affects their planarity. In accordance with the present invention this is accomplished by loading each member into a three-point-suspension cassette unit by means of a fixture that performs the loading in a precise and essentially stress-free manner.

Structure and bonding of 4,4-dicyano-2,3-diphenyltriafulvene

Abstract: The crystals of the title compound are monoclinic, C2/c, with a= 14.31(1), b= 10.100(8), c= 9.607(8)A, β= 106.57(6)°, and Z= 4. Least-squares refinement, based on 640 independent reflections, converged at R 0.043. The molecules possess crystallographic C2 symmetry, with slight deviations from planarity. A simple electronic scheme, based on the Walsh model and a HMO calculation, for this and other similar cyclopropyl compounds, is proposed to rationalize the bond length values. This model also suggests that the high value of the dipole moment in this compound may be due to a partial delocalization of the electron density charge over the ring.

Solutions of simple dual bootstrap models satisfying Lee--Veneziano relation and the smallness of cut discontinuities

Abstract: To investigate the t-dependent solutions of simple dual bootstrap models, two general formulations are discussed, one without and one with cut cancellation at the planar level. The possible corresponding production mechanisms are discussed. In contrast to Bishari's formulation, both models recover the Lee-Veneziano relation, i.e., in the peak approximation the Pomeron intercept is unity. The solutions based on an exponential form for the reduced triple-Reggeon vertex for both models are discussed in detail. Also calculated are the cut discontinuities for both models and for Bishari's and it is shown that at both the planar and cylinder levels they are small compared with the corresponding pole residues. Precocious asymptotic planarity is also found in the solutions.