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Stable Set (stable + set)
Selected AbstractsA structure theorem for graphs with no cycle with a unique chord and its consequencesJOURNAL OF GRAPH THEORY, Issue 1 2010Nicolas Trotignon Abstract We give a structural description of the class ,, of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in ,, is either in some simple basic class or has a decomposition. Basic classes are chordless cycles, cliques, bipartite graphs with one side containing only nodes of degree 2 and induced subgraphs of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for ,,, i.e. every graph in ,, can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations, and all graphs built this way are in ,,. This has several consequences: an ,,(nm) -time algorithm to decide whether a graph is in ,,, an ,,(n+ m) -time algorithm that finds a maximum clique of any graph in ,,, and an ,,(nm) -time coloring algorithm for graphs in ,,. We prove that every graph in ,, is either 3-colorable or has a coloring with , colors where , is the size of a largest clique. The problem of finding a maximum stable set for a graph in ,, is known to be NP-hard. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 31,67, 2010 [source] Global existence and uniform stability of solutions for a quasilinear viscoelastic problemMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 6 2007Salim A. Messaoudi Abstract In this paper the nonlinear viscoelastic wave equation in canonical form with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set. Copyright © 2006 John Wiley & Sons, Ltd. [source] Some remarks on global existence to the Cauchy problem of the wave equation with nonlinear dissipationMATHEMATISCHE NACHRICHTEN, Issue 12 2008Nour-Eddine Amroun Abstract In this paper we prove the existence of global decaying H2 solutions to the Cauchy problem for a wave equation with a nonlinear dissipative term by constructing a stable set in H1(,n). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] On d -threshold graphs and d -dimensional bin packingNETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2004Alberto Caprara Abstract We illustrate efficient algorithms to find a maximum stable set and a maximum matching in a graph with n nodes given by the edge union of d threshold graphs on the same node set, in case the d graphs in the union are known. Actually, because the edge set of a threshold graph can be implicitly represented by assigning values to the nodes, we assume that we know these values for each of the d graphs in the union. We present an O(n log n + nd,1) time algorithm to find a maximum stable set and an O(n2) time algorithm to find a maximum matching, in case d is constant. For the case d = 2, the running time of the latter is reduced to O(n log n) provided an additional technical condition is satisfied. The natural application of our results is the fast computation of lower bounds for the d -dimensional bin packing problem, for which the compatibility relations between items are represented by the edge union of d threshold graphs with one node for each item, the value of the node for the i -th graph being equal to the size of the item on the i -th dimension. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(4), 266,280 2004 [source] | ||||||||||