Bridge Graphs Research Articles

A Helly theorem for geodesic convexity in strongly dismantlable graphs

A (finite or infinite) graph G is strongly dismantlable if its vertices can be linearly ordered x 0,…, x α so that, for each ordinal β < α, there exists a strictly increasing finite sequence ( i j ) 0 ⩽ j ⩽ n of ordinals such that i 0 = β, i n = α and x i j+1 is adjacent with x i j and with all neighbors of x i j in the subgraph of G induced by { x y : β ⩽ γ ⩽ α}. We show that the Helly number for the geodesic convexity of such a graph equals its clique number. This generalizes a result of Bandelt and Mulder (1990) for dismantlable graphs. We also get an analogous equality dealing with infinite families of convex sets.

A matrix characterization of induced paths in bridge graphs

We present a matrix characterization of induced paths in bridge graphs. This matrix characterization is used to prove the existence of a canonical form for induced paths. Matrices which display this canonical form have only entries of 0,1, and 2. It is then shown that an induced path P=(B 1,B 2,…B q),where Q≥3, can be made into an induced cycle with the addition of a bridge if and only if the canonical form matrices for the path P and the reverse path P R are equivalent and of the form … .

An $O(m\log n)$-Time Algorithm for the Maximal Planar Subgraph Problem

Based on a new version of the Hopcroft and Tarjan planarity testing algorithm, this paper develops an $O(m\log n)$-time algorithm to find a maximal planar subgraph.

Canonical forms for cycles in bridge graphs

Let G = (V,E) be a biconnected graph and let C be a cycle in G. The subgraphs of G identified with the biconnected components of the contraction of C in G are called the bridges of C. Associated with the set of bridges of a cycle C is an auxilliary graphical structure GC called a bridge graph or an overlap graph. Such auxilliary graphs have provided important insights in classical graph theory, algorithmic graph theory, and complexity theory. In this paper, we use techniques from algorithmic combinatorics and complexity theory to derive canonical forms for cycles in bridge graphs. These canonical forms clarify the relationship between cycles in bridge graphs, the structure of the underlying graph G, and lexicographic order relations on the vertices of attachment of bridges of a cycle. The first canonical form deals with the structure of induced bridge graph cycles of length greater than three. Cycles of length three in bridge graphs are studied from a different point of view, namely that of the characteri...

Highly asymmetric electrolytes: beyond the hypernetted chain integral equation

Corrections to the hypernettede chain (HNC) integral equation, based on an approximation for the sum of bridge graphs, have been evaluated for a model electrolyte solution asymmetric in both charge and size. The system was previously studied by the HNC integral equation and by the Monte Carlo simulation technique. The approximations for the sum of the bridge graphs are used, together with the Ornstein-Zernike equation, to obtain new pair correlation functions

On diameters and radii of bridged graphs

A graph G is bridged if it contains no isometric cycles of length greater than 3. For each positive integer p, let T p be the graph obtained by triangulating an equilateral triangle of side p with equilateral triangles of side 1. It is shown that the radius and diameter of a bridged graph containing no isometric T p satisfy the inequality 6 r⩽3 d+ p+3. Thus, for each fixed p, the radius of a bridged graph containing no isometric T p is within a constant of its theoretical lower bound. Also, letting p= d+1, it follows that the radius and diameterof an arbitrary bridged graph satisfy the inequality 3 r⩽2 d+2. The graphs T p , p⩾1, show that this bound on the radius is best possible. Two results of Chang and Nemhauser concerning diameters and radii of chordal graphs are also corollaries.

On bridged graphs and cop-win graphs

A subgraph H of a graph G is isometric if the distance between any pair of vertices in H is the same as that in G. A graph is bridged if it contains no isometric cycles of length greater than three (roughly, each cycle of length greater than three has a shortcut). It is proved that every nontrivial bridged graph has a pair of adjacent vertices u, v, with v adjacent to everything to which u is adjacent. (This result was conjectured by P. Hell, and independently raised as a question, though in a different form, by R. E. Jamison). From this, it follows that every bridged graph G has a vertex u such that G − u is an isometric subgraph, and hence also bridged. The latter is a result of Farber. It is also proved that a connected graph is bridged if and only if every isometric subgraph is a cop-win graph, as defined by Nowakowski and Winkler.

On the mechanical instability of hard-sphere systems

The mechanical instability condition for the hard-sphere fluid is analysed by considering many different approximations to the direct correlation function. These have been obtained by adding to the truncated density expansion of the Percus-Yevick solution the contributions due to the 2nd and 3rd order parallel and bridge graphs. One finds that the bifurcation density values are smaller than the close-packing value (√2) and, depending on the truncation order, they vary in the range [0·97, 1·36].

Corrections to the HNC equation for associating electrolytes

The accuracy of the HNC approximation and modifications based on a simple approximation for the sum of bridge graphs has been studied for a charged soft sphere model for dilute aqueous 2-2 electrolytes. The accuracy is judged based on extensive Monte Carlo (MC) simulation for three representative concentrations: 0.005, 0.0625, and 0.2 M. The MC simulations find monotonically increasing like-charge correlation functions in agreement with the modified HNC results but in contrast with the HNC prediction of a local maximum at low concentration. Nevertheless, HNC provides the best overall quantitative match with MC except at very low concentration; at 5 mM the modifications are a significant improvement. For unlike charge correlations, there is good agreement among all methods; at the ionic contact peak, however, HNC underestimates the peak height and the modifications provide improved accuracy.

Systematic graphical correction of the hypernetted chain theory for the one-component plasma

The Fourier transforms of the lowest two orders of bridge graphs in the Abe-Meeron expansion for the one-component plasma are evaluated numerically after using the Goursat-Feynman relations to reduce the dimensionality of the integrals involved. After inverse Fourier transformation, the results are used to form approximants for the complete set of bridge graphs. These are then incorporated into modified hypernetted chain (HNC) theories which are expected to be useful in the intermediate-coupling regime. The results of the modified HNC equations improve upon the usual HNC results in terms of virial-compressibility consistency and they suggest that the excess energies found by Monte Carlo simulation may not be very reliable in the intermediate-coupling regime.

A semi-empirical modified hypernetted chain equation for the one-component plasma

The hypernetted chain (HNC) equation for the one-component plasma is corrected by including a single-parameter expression for the bridge graphs whose functional form ensures that the correct long-wavelength limit of the HNC results is retained. The parameter is chosen to improve the short-range behaviour.

Exact calculations of Coulomb bridge graphs in one dimension

Bridge nodal diagrams built upon the screened Debye interaction are considered in 2+ epsilon dimensions for the classical one-component plasma (OCP) model. Their asymptotic behaviour is shown to be nearly epsilon -independent. They are evaluated analytically in one dimension. The corresponding asymptotic decay approximately exp(-Kkr), with Kk a positive integer >or=2, is thus extrapolated to any epsilon .

Asymptotic behavior of equilibrium pair correlations in a dense electron gas

The nodal expansion of the potential of average force is worked out systematically by iterating all the nonconvolution compact graphs from the order at which they appear first to infinity. For this purpose, we perform an exhaustive study of the asymptotic behavior of the bridge (without articulation points) graphs, and show that they decrease as $\frac{\ensuremath{\beta}{e}^{\ensuremath{-}\ensuremath{\alpha}r}}{r}$, with $\ensuremath{\alpha}&gt;1$ and $r$ in units of the Debye length ${\ensuremath{\lambda}}_{D}$. We demonstrate that the asymptotic ${w}_{2}(r)$ expansion is obtained from the resummation of the longest convolution chains [$l\ensuremath{\ge}2(n\ensuremath{-}1)$] with $n\ensuremath{-}1$ two-bubbles and $0\ensuremath{\le}c\ensuremath{\le}n$ Debye lines, which allow for a systematic improvement of the usual hypernetted-chain (HNC) approximation with the replacement of one, two, or more two-bubbles by bridge graphs. Substantial simplification of the final expression is achieved with the aid of the $n$-bubble sum which decreases asymptotically faster than the Debye line. The onset of short-range order is shown to arise at the critical value ${\ensuremath{\Lambda}}_{c}=4.247$ of the plasma parameter $\ensuremath{\Lambda}=\frac{{e}^{2}}{{k}_{B}}T{\ensuremath{\lambda}}_{D}$ in excellent agreement with the Del Rio-DeWitt calculations.

Asymptotic behaviour of the equilibrium pair correlation in the classical electron gas

The non-convolution graphs, sometimes referred to as bridge graphs, which figure in the Mayer-Salpeter expansion of the potential of average forcew2(r) with respect to the plasma parameter, are systematically studied. They decrease at infinity faster than the first-order Debye graph e-r/r. An exact expression for w2(r) which allows a systematic improvement of the hyper-netted chain (HNC) approximation is obtained.

Jordan circuits of a graph

Let G be connected, finite graph. Let C be a circuit of G. β( C), the strong bridge graph of C in G, is defined as follows: (1) the vertices of β( C) are the bridges of C in G, and (2) there is an edge in β( C) joining a pair of vertices B 1 and B 2 if and only if B 1 and B 2 separate each other relative C. • Theorem. Let G be a finite, connected graph. G is planar if and only if β(C) is bipartite for each circuit C in G. • Lemma. Let G be a finite, connected graph. G is not planar if and only if there is a circuit C of G for which β(C) contains a loop or a triangle. This Lemma isolates the crucial step in a new proof of the Kuratowski Theorem.

Graph-theoretic analysis of the BGY theory of classical fluids with application to the fourth-virial coefficient for the 12-6 potential

The Born-Green-Yvon (BGY) approximate theory relating the radial distribution function g(r) for a classical fluid to the pair potential of intermolecular force is analysed in terms of two-rooted graphs with Mayer f functions as links. We find that the BGY theory neglects many terms in the gradient of the sum of all simple graphs, and retains only those terms obtained from bridge graphs with an f link between root point 1 and the first nodal point by replacing the f link with its gradient. Each term in the density expansion of the gradient of g(r) may be integrated so that the BGY theory may be compared with the exact, Percus-Yevick, and hypernetted chain theories at any order of density. As an application of this analysis, the fourth-virial coefficient as a function of temperature is calculated in the BGY approximation for the Lennard-Jones 12-6 potential.