5 research outputs found

    Residual reliability of P-threshold graphs

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    We solve the problem of computing the residual reliability (the RES problem) for all classes of P-threshold graphs for which efficient structural characterizations based on decomposition to indecomposable components have been established. In particular, we give a constructive proof of existence of linear algorithms for computing residual reliability of pseudodomishold, domishold, matrogenic and matroidal graphs. On the other hand, we show that the RES problem is #P-complete on the class of biregular graphs, which implies the #P-completeness of the RES problem on the classes of indecomposable box-threshold and pseudothreshold graph

    HH-product and HH-threshold graphs

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    This paper is the continuation of the research of the author and his colleagues of the {\it canonical} decomposition of graphs. The idea of the canonical decomposition is to define the binary operation on the set of graphs and to represent the graph under study as a product of prime elements with respect to this operation. We consider the graph together with the arbitrary partition of its vertex set into nn subsets (nn-partitioned graph). On the set of nn-partitioned graphs distinguished up to isomorphism we consider the binary algebraic operation H\circ_H (HH-product of graphs), determined by the digraph HH. It is proved, that every operation H\circ_H defines the unique factorization as a product of prime factors. We define HH-threshold graphs as graphs, which could be represented as the product H\circ_{H} of one-vertex factors, and the threshold-width of the graph GG as the minimum size of HH such, that GG is HH-threshold. HH-threshold graphs generalize the classes of threshold graphs and difference graphs and extend their properties. We show, that the threshold-width is defined for all graphs, and give the characterization of graphs with fixed threshold-width. We study in detail the graphs with threshold-widths 1 and 2

    Subject Index Volumes 1–200

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    Pseudodomishold graphs

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    Residual reliability of P-threshold graphs

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    AbstractWe solve the problem of computing the residual reliability (the RES problem) for all classes of P-threshold graphs for which efficient structural characterizations based on decomposition to indecomposable components have been established. In particular, we give a constructive proof of existence of linear algorithms for computing residual reliability of pseudodomishold, domishold, matrogenic and matroidal graphs. On the other hand, we show that the RES problem is #P-complete on the class of biregular graphs, which implies the #P-completeness of the RES problem on the classes of indecomposable box-threshold and pseudothreshold graphs
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