Abstract
The reliability polynomial of a graph gives the probability that a graph remains operational when all its edges could fail independently with a certain fixed probability. In general, the problem of finding uniformly most reliable graphs inside a family of graphs, that is, one graph whose reliability is at least as large as any other graph inside the family, is very difficult. In this paper, we study this problem in the family of graphs containing a hamiltonian cycle.
Highlights
Notation and terminologyIn the reliability context, networks are modeled by graphs
In this paper uniformly most reliable hamiltonian graphs have been characterized for m ≤ n + 2 and we give some light for the case m = n + 3 which somehow follow the previous cases
The situation is different for m ≥ n + 4, where the problem is totally open except for some small cases found by computer
Summary
Networks are modeled by graphs. We recall that a graph is an ordered pair G = (V, E), where V is a non empty set of nodes or vertices, and E is a set of unordered pairs of different elements of V , called links or edges. A main problem in the reliability context is concerned with the design of networks with ‘high’ reliability To this end, let G(n, m) be the set of all simple connected graphs with n vertices and m edges. Every uniformly most reliable graph in the case m = n + 1 (n ≥ 5) is constructed taking a graph of order 5 as a basis and subdividing edges and adding vertices one by one by following the sequence A, B, C, A, B, C, .
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