Invulnerability of planar two-tree networks

Let us model the network by an undirected graph G, in which each edge fails independently with probability q ∈ [ 0 , 1 ] , the all-terminal reliability of the network is the probability that the graph G remains connected and it is one of the important measure for the invulnerability of the network. Clearly, the all-terminal reliability can be written as a polynomial in either q. Theoretically, it has been proved that calculating all-terminal reliability for planar networks is #P-complete. Up now we cannot find any results for some special classes of the planar networks. This paper studies the all-terminal reliability of the two-tree networks and its generalizations. A linear algorithm for calculating the reliability of a planar two-tree networks is proposed, with algorithmic complexity O ( n ) , where n is the number of steps. Then both the uniformly-most reliable network model and the uniformly-worst reliable network model are determined by using the algorithm. Finally, a polynomial algorithm for planar two-connected networks is proposed based on an expended algorithm and the corresponding uniformly-most reliable network model and uniformly-worst reliable network model are also presented, with computational complexity O ( n ) , where n is the number of steps.