Nauka i Obrazovanie (Jan 2016)
Completely Described Undirected Graph Structure
Abstract
The objects of research are undirected graphs. The paper considers a problem of their isomorphism. A literature analysis of its solution, has shown that there is no way to define a complete graph invariant in the form of unique structural characteristics of each its vertex, which has a computational complexity of definition better than О (n 4 ).The work objective is to provide the characteristics of the graph structure, which could be used to solve the problem of their isomorphism for a time better than О (n 4 ). As such characteristics, the paper proposes to use the set of codes of tree roots of all the shortest - in terms of the number of edges - paths from each vertex to the others, uniquely defining the structure of each tree. It proves the theorem that it is possible to reduce the problem of isomorphism of the undirected graphs to the isomorphism problem of their splitting into the trees of all the shortest - in terms of the number of edges - paths of each vertex to the others. An algorithm to construct the shortest paths from each vertex to all others and to compute codes of their vertices has been developed. As the latter, are used Aho-codes, which find application in recognising the isomorphism of trees. The computational complexity to obtain structural characteristics of vertices has been estimated to be about О (n 3 ).The pilot studies involved the full-scale experiment using the developed complex programmes to generate raw data, i.e. analytic representation of the graph with the number of vertices equal to 1200, and a programme to provide codes of the tree roots. To have an estimate of - "the worst" in terms of time - complexity of expansion algorithm of graphs into trees of the shortest paths and define the codes of their roots has been an experimentally studied how the number of tree vertices depends on the graph density. For the worst case was obtained a dependence of the number of tree vertices on the number of graph vertices, approximated by the quadratic function, which is consistent with the theoretical estimates. The experimental results of studies, in general, prove those of theoretical estimates.
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