Graph Self-Transformation Model Based on the Operation of Change the End of the Edge

We consider a distributed network whose topology is described by an undirected graph. The network itself can change its topology, using special “commands” provided by its nodes. The work proposes an extremely local atomic transformation acb of a change the end c of the edge ac, “moving” along the edge cb from vertex c to vertex b. As a result of this operation, the edge ac is removed, and the edge ab is added. Such a transformation is performed by a “command” from a common vertex c of two adjacent edges ac and cb. It is shown that from any tree you can get any other tree with the same set of vertices using only atomic transformations. If the degrees of the tree vertices are bounded by the number d (d 3), then the transformation does not violate this restriction. As an example of the purpose of such a transformation, the problems of maximizing and minimizing the Wiener index of a tree with a limited degree of vertices without changing the set of its vertices are considered. The Wiener index is the sum of pairwise distances between the vertices of a graph. The maximum Wiener index has a linear tree (a tree with two leaf vertices). For a root tree with a minimum Wiener index, its type and method for calculating the number of vertices in the branches of the neighbors of the root are determined. Two distributed algorithms are proposed: transforming a tree into a linear tree and transforming a linear tree into a tree with a minimum Wiener index. It is proved that both algorithms have complexity no higher than 2n–2, where n is the number of tree vertices. We also consider the transformation of arbitrary undirected graphs, in which there can be cycles, multiple edges and loops, without restricting the degree of the vertices. It is shown that any connected graph with n vertices can be transformed into any other connected graph with k vertices and the same number of edges in no more than 2(n+k)–2.