An efficient algorithm to recognize locally equivalent graphs

To locally complement a simple graphF at one of its verticesv is to replace the subgraph induced byF onn(v)={w:vw is an edge ofF} by the complementary subgraph. Graphs related by a sequence of local complementations are said to be locally equivalent. We associate a system of equations with unknowns inGF(2) to any pair of graphs {F, F}, so thatF is locally equivalent toF if and only if the system has a solution. The equations are either linear and homogenous or bilinear, and we find a solution, if any, in polynomial time.With partial support of P. R. C. Mathématiques et Informatique.     

An efficient algorithm for finding the largest chain graph according to a given chain graph

Chain graph (CG) is a general model of graphical Markov models. Some different chain graphs may describe the same conditional independence structure, then we say that these CGs are Markov equivalent. In 1990 Frydenberg showed that every class of Markov equivalent CGs has a CG which is called the largest chain graph with the greatest number of lines. This paper presents an efficient algorithm for finding the largest chain graph of the corresponding Markov equivalent class of a given CG. The computational complexity of the algorithm is O(n3). It is more efficient than the complexity O(n!) of the present algorithms. Also a more intuitive graphical characterization of the largest chain graph is provided based on the algorithm in this paper.     

An exact algorithm for graph partitioning

An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate. It is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set. The concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function Hessian, or by a diagonal shift obtained by solving a semidefinite programming problem. Numerical results show that the proposed algorithm is competitive with state-of-the-art graph partitioning codes.     

An algorithm for the determination of longest distances in a graph

This paper presents an algorithm for ranking the vertices of a directed graph. Its space and time requirements are bounded byc ₁ n ² +c ₂, wheren is the number of vertices of the graph andc ₁,c ₂ are positive constants which are independent of the size or other properties of the graph.The algorithm can be easily modified to solve the problem of determining longest distances from a vertex to all other vertices in a positive real valued graph with at mostc ₁ n ² +c ₂ elementary operations; the same result holds for shortest distances in negative real valued graphs.     

An algorithm for calculating the independence and vertex-cover polynomials of a graph

The independence polynomial, ω(G,x)=∑w_kx^k, of a graph, G, has coefficients, w_k, that enumerate the ways of selecting k vertices from G so that no two selected vertices share an edge. The independence number of G is the largest value of k for which w_k≠0. Little is known of less straightforward relationships between graph structure and the properties of ω(G,x), in part because of the difficulty of calculating values of w_k for specific graphs. This study presents a new algorithm for these calculations which is both faster than existing ones and easily adaptable to high-level computer languages.     

All shortest distances in a graph. An improvement to Dantzig's inductive algorithm

Dantzig's inductive algorithm is one of the best solutions to find all shortest distances in a graph. A change in the order of computations makes it possible to improve that algorithm by exploiting nonexistent arcs.     

An efficient memetic algorithm for the graph partitioning problem

We consider the component testing problem of a device that is designed to perform a mission consisting of a random sequence of phases with random durations. Testing is done at the component level to attain desired levels of mission reliability at minimum cost. The components fail exponentially where the failure rate depends on the phase of the mission. The reliability structure of the device involves a series connection of nonidentical components with different failure characteristics. The optimal component testing problem is formulated as a semi-infinite linear program. We present an algorithmic procedure to compute optimal test times based on the column generation technique, and illustrate it with numerical examples.     

An adaptive multi-start graph partitioning algorithm for structuring cellular networks

In mobile network design, the problem of assigning network elements to controllers when defining network structure can be modeled as a graph partitioning problem. In this paper, a comprehensive analysis of a sophisticated graph partitioning algorithm for grouping base stations into packet control units in a mobile network is presented. The proposed algorithm combines multi-level and adaptive multi-start schemes to obtain high quality solutions efficiently. Performance assessment is carried out on a set of problem instances built from measurements in a live network. Overall results confirm that the proposed algorithm finds solutions better than those obtained by the classical multi-level approaches and much faster than classical multi-start approaches. The analysis of the optimization surface shows that the best local minima values follow a Gumbel distribution, which justifies the stagnation of naive multi-start approaches after a few attempts. Likewise, the analysis shows that the best local minima share strong similarities, which is the reason for the superiority of adaptive multi-start approaches. Finally, a sensitivity analysis shows the best internal parameter settings in the algorithm.     

An efficient condition for a graph to be Hamiltonian

Let G=(V,E) be a 2-connected simple graph and let d_G(u,v) denote the distance between two vertices u,v in G. In this paper, it is proved: if the inequality d_G(u)+d_G(v)?|V(G)|-1 holds for each pair of vertices u and v with d_G(u,v)=2, then G is Hamiltonian, unless G belongs to an exceptional class of graphs. The latter class is described in this paper. Our result implies the theorem of Ore [Note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55]. However, it is not included in the theorem of Fan [New sufficient conditions for cycles in graph, J. Combin. Theory Ser. B 37 (1984) 221-227].     

Worst case analysis of a graph coloring algorithm

In this paper, we show that a graph coloring heuristic developed by Brélaz may use n colors to color a 3-colorable graph with O(n) vertices.     

An algorithm for determining the automorphism partitioning of an undirected graph

An Algol 60 program for determining the automorphism partitioning of an undirected unlabelled graph is presented. For graphs that do not contain strongly regular subgraphs, the time is at worstO(n ⁴) wheren is the number of vertices in the graph. The algorithm is based on the Corneil-Gotlieb conjecture.     

An example of a superproportional graph

In the theory of random graphs, several classes of graphs occur that are exceptional cases in limit theorems for subgraph counts. The purpose of this article is to show the existence of graphs in one of these classes, by providing an explicit, computer generated, example. We also show that the class is closed under complementation. © 1994 John Wiley & Sons, Inc.     

A non-factorial algorithm for canonical numbering of a graph

A graph certificate or canonical form is a short unique (up to isomorphism) representation of the graph. Thus two graphs are isomorphic iff their certificates are identical. In this paper an O(cⁿ) graph isomorphism algorithm which also yields a certificate of the graph is presented. The certificate produced by this algorithm is a canonical numbering of the vertices of the graph.     

An invariant theory approach to graph enumeration

The paper exploits a connection between the graphs on n vertices and the invariants of the pair group S⁽²⁾_n.     

AN APPROACH TO CONSTRUCT AN END-REGULAR GRAPH

In this paper,an approach to construct a nontrivial end-regular graph of any order under some conditions are given.     

An iterative approach to graph irregularity strength

An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we take an iterative approach to calculating the irregularity strength of a graph. In particular, we develop a new algorithm that determines the exact value s(T) for trees T in which every two vertices of degree not equal to two are at distance at least eight.