First,the state space tree method for finding communication network overall re-liability is presented.It directly generates one disjoint tree multilevel polynomial of a networkgraph.Its advantages are smaller computat...First,the state space tree method for finding communication network overall re-liability is presented.It directly generates one disjoint tree multilevel polynomial of a networkgraph.Its advantages are smaller computational effort(its computing time complexity is O(en_l),where e is the number of edges and n_l is the number of leaves)and shorter resulting expression.Second,based on it an exact decomposition algorithm for finding communication network overallreliability is presented by applying the hypergraph theory.If we use it to carry out the m-timedecomposition of a network graph,the communication network scale which can be analyzed by acomputer can be extended to m-fold.展开更多
A path decomposition of a graph G is a list of paths such that each edge appears in exactly one path in the list. G is said to admit a {Pl }-decomposition if G can be decomposed into some copies of Pl, where Pl is a p...A path decomposition of a graph G is a list of paths such that each edge appears in exactly one path in the list. G is said to admit a {Pl }-decomposition if G can be decomposed into some copies of Pl, where Pl is a path of length l - 1. Similarly, G is said to admit a {Pl, Pk}-decomposition if G can be decomposed into some copies of Pl or Pk. An k-cycle, denoted by Ck, is a cycle with k vertices. An odd tree is a tree of which all vertices have odd degree. In this paper, it is shown that a connected graph G admits a {P3, P4}-decomposition if and only if G is neither a 3-cycle nor an odd tree. This result includes the related result of Yan, Xu and Mutu. Moreover, two polynomial algorithms are given to find {P3}-decomposition and {P3, P4}-decomposition of graphs, respectively. Hence, {P3}-decomposition problem and {P3, P4}-decomposition problem of graphs are solved completely.展开更多
According to the researches on theoretic basis in part Ⅰ of the paper, the spanning tree algorithms solving the maximum independent set both in even network and in odd network have been developed in this part, part ...According to the researches on theoretic basis in part Ⅰ of the paper, the spanning tree algorithms solving the maximum independent set both in even network and in odd network have been developed in this part, part Ⅱ of the paper. The algorithms transform first the general network into the pair sets network, and then decompose the pair sets network into a series of pair subsets by use of the characteristic of maximum flow passing through the pair sets network. As for the even network, the algorithm requires only one time of transformation and decomposition, the maximum independent set can be gained without any iteration processes, and the time complexity of the algorithm is within the bound of O(V3). However, as for the odd network, the algorithm consists of two stages. In the first stage, the general odd network is transformed and decomposed into the pseudo-negative envelope graphs and generalized reverse pseudo-negative envelope graphs alternately distributed at first; then the algorithm turns to the second stage, searching for the negative envelope graphs within the pseudo-negative envelope graphs only. Each time as a negative envelope graph has been found, renew the pair sets network by iteration at once, and then turn back to the first stage. So both stages form a circulation process up to the optimum. Two available methods, the adjusting search and the picking-off search are specially developed to deal with the problems resulted from the odd network. Both of them link up with each other harmoniously and are embedded together in the algorithm. Analysis and study indicate that the time complexity of this algorithm is within the bound of O(V5).展开更多
文摘First,the state space tree method for finding communication network overall re-liability is presented.It directly generates one disjoint tree multilevel polynomial of a networkgraph.Its advantages are smaller computational effort(its computing time complexity is O(en_l),where e is the number of edges and n_l is the number of leaves)and shorter resulting expression.Second,based on it an exact decomposition algorithm for finding communication network overallreliability is presented by applying the hypergraph theory.If we use it to carry out the m-timedecomposition of a network graph,the communication network scale which can be analyzed by acomputer can be extended to m-fold.
基金Supported by the National Natural Science Foundation of China(No.10301010)Science and Technology Commission of Shanghai Municipality(No.04JC14031)Natural Science Project of Chuzhou University(No.2006kyy017)
文摘A path decomposition of a graph G is a list of paths such that each edge appears in exactly one path in the list. G is said to admit a {Pl }-decomposition if G can be decomposed into some copies of Pl, where Pl is a path of length l - 1. Similarly, G is said to admit a {Pl, Pk}-decomposition if G can be decomposed into some copies of Pl or Pk. An k-cycle, denoted by Ck, is a cycle with k vertices. An odd tree is a tree of which all vertices have odd degree. In this paper, it is shown that a connected graph G admits a {P3, P4}-decomposition if and only if G is neither a 3-cycle nor an odd tree. This result includes the related result of Yan, Xu and Mutu. Moreover, two polynomial algorithms are given to find {P3}-decomposition and {P3, P4}-decomposition of graphs, respectively. Hence, {P3}-decomposition problem and {P3, P4}-decomposition problem of graphs are solved completely.
文摘According to the researches on theoretic basis in part Ⅰ of the paper, the spanning tree algorithms solving the maximum independent set both in even network and in odd network have been developed in this part, part Ⅱ of the paper. The algorithms transform first the general network into the pair sets network, and then decompose the pair sets network into a series of pair subsets by use of the characteristic of maximum flow passing through the pair sets network. As for the even network, the algorithm requires only one time of transformation and decomposition, the maximum independent set can be gained without any iteration processes, and the time complexity of the algorithm is within the bound of O(V3). However, as for the odd network, the algorithm consists of two stages. In the first stage, the general odd network is transformed and decomposed into the pseudo-negative envelope graphs and generalized reverse pseudo-negative envelope graphs alternately distributed at first; then the algorithm turns to the second stage, searching for the negative envelope graphs within the pseudo-negative envelope graphs only. Each time as a negative envelope graph has been found, renew the pair sets network by iteration at once, and then turn back to the first stage. So both stages form a circulation process up to the optimum. Two available methods, the adjusting search and the picking-off search are specially developed to deal with the problems resulted from the odd network. Both of them link up with each other harmoniously and are embedded together in the algorithm. Analysis and study indicate that the time complexity of this algorithm is within the bound of O(V5).
