About one problem of analysis of the topology of communication networks
Abstract
The paper discusses heuristic nonlinear algorithms for checking 2-multiple edge connectivity. This is the term we introduce, a graph has 2-multiple edge connectivity in the case when there are at least 2 edge-independent paths between any two of its vertices; it is important to note that this concept does not coincide with the concept of edge $2$-connectivity of a graph, which is more studied in the literature.
For the last problem, polynomial algorithms have been previously described in the literature, and their small modifications also give algorithms for solving "our" problem. However, we do not consider them: they are quite complex to implement, and we need algorithms in which, at the request of the customer, it is often necessary to make some modifications during its execution.
In the problems considered in this paper, we use such auxiliary algorithms that can be called heuristic; as a result, the full versions of the algorithms under consideration can also be called heuristic. In the paper we consider 3 different groups of algorithms designed to test 2-multiple edge connectivity.
Namely, first we give explicit algorithms for such verification, i.e. algorithms in which the necessary paths are constructed explicitly. Next, in the following algorithm, we check the possible existence of a common cycle for any two vertices of a given graph. The third algorithm is that we check 2-multiple edge connectivity based, firstly, on the absence of bridges and, secondly, on the presence of at least some cycle involving it for any vertex of the graph.
We present a scheme for proving the equivalence of these three algorithms, and after that, a brief description of computational experiments.
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