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1. FAULT-TOLERANT MAXIMAL LOCAL-CONNECTIVITY ON CAYLEY GRAPHS GENERATED BY TRANSPOSITION TREES.

Author

SHIH, LUN-MIN, CHIANG, CHIEH-FENG, HSU, LIH-HSING, and TAN, JIMMY J. M.

Subjects
*PAPER, *VERTEX operator algebras, *GRAPH connectivity, *CAYLEY graphs, *GRAPH theory
Abstract

The local connectivity of two vertices is defined as the maximum number of internally vertex-disjoint paths between them. In this paper, we define two vertices as maximally local-connected, if the maximum number of internally vertex-disjoint paths between them equals the minimum degree of these two vertices. Moreover, we show that an (n-1)-regular Cayley graph generated by transposition tree is maximally local-connected, even if there are at most (n-3) faulty vertices in it, and prove that it is also (n-1)-fault-tolerant one-to-many maximally local-connected. [ABSTRACT FROM AUTHOR]

Published
2009
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2. Multicyclic treelike reflexive graphs

Author

Radosavljević, Z. and Rašajski, M.

Subjects
*GRAPH theory, *SUCCULENT plants, *CYCLES, *PAPER
Abstract

Abstract: A simple graph is reflexive if its second largest eigenvalue does not exceed 2. A graph is treelike (sometimes also called a cactus) if all its cycles (circuits) are mutually edge-disjoint. In a lot of cases one can establish whether a given graph is reflexive by identifying and removing a single cut-vertex (Theorem 1). In this paper we prove that, if this theorem cannot be applied to a connected treelike reflexive graph G and if all its cycles do not have a common vertex (do not form a bundle), such a graph has at most five cycles (Theorem 2). On the same conditions, in Theorem 3 we find all maximal treelike reflexive graphs with four and five cycles. [Copyright &y& Elsevier]

Published
2005
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3. A new algorithm for the maximum weighted stable set problem in claw-free graphs

Author

Gianpaolo Oriolo, Ugo Pietropaoli, Gautier Stauffer, Dipartimento di Ingegneria dell'Impresa [Roma], Università degli Studi di Roma Tor Vergata [Roma], Reformulations based algorithms for Combinatorial Optimization (Realopt), Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS)-Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Combinatorial optimization, Computer science, 0211 other engineering and technologies, 02 engineering and technology, Polynomials, 01 natural sciences, Domain decomposition methods, Scheduling algorithms, Matching problems, Chordal graph, Polynomial-time algorithms, Longest path, ComputingMilieux_MISCELLANEOUS, decomposition theorems, 021103 operations research, Quasi-line graphs, Mathematical programming, Integer programming, Longest path problem, Graph, Modular decomposition, 010201 computation theory & mathematics, (algorithmic) complexity, Auxiliary graphs, Claw-free graphs, Mathematical transformations, Set theory, Settore MAT/09 - Ricerca Operativa, Algorithm, Algorithms, Optimization, Paper, Aerospace applications, Boolean functions, Combinatorial mathematics, Curve fitting, Dynamic programming, Evolutionary algorithms, Farm buildings, Graph theory, Linear programming, Nonlinear programming, Polynomial approximation, Reduction, Trees (mathematics), General (CO), graph reductions, international conferences, New algorithm, Polynomial-time, Reduction tools, Running in, time bound, 0102 computer and information sciences, Combinatorics, Indifference graph, Pathwidth, Cograph, [INFO]Computer Science [cs], Graph coloring, Time complexity, Discrete mathematics, 1-planar graph, Vertex (geometry), Independent set
Abstract

In this paper, we introduce two powerful graph reductions for the maximum weighted stable set (mwss) in general graphs. We show that these reductions allow to reduce the mwss in claw-free graphs to the mwss in a class of quasi-line graphs, that we call bipolar-free. For this latter class, we provide a new algorithmic decomposition theorem running in polynomial time. We then exploit this decomposition result and our reduction tools again to transform the problem to either a single matching problem or a longest path computation in an acyclic auxiliary graph (in this latter part we use some results of Pulleyblank and Shepherd [10]). Putting all the pieces together, the main contribution of this paper is a new polynomial time algorithm for the mwss in claw-free graphs. A rough analysis of the complexity of this algorithm gives a time bound of O(n6), where n is the number of vertices in the graph, and which we hope can be improved by a finer analysis. Incidentally, we prove that the mwss problem can be solved efficiently for any class of graphs that admits a "suitable" decomposition into pieces where the mwss is easy.

Published
2008