Minimal joint graph structure synthesis

The goal of this work is to research the structure of minimal joint graph and its properties.
The new concepts are given: backbone connectivity, significant narrowing, demesne vertex, vassal, homage, feud, sinew, bunch, monoclique, stereoclique, obligatory edges graph. Particularly, a minimal joint graph of maximal knowledge patterns was defined as the a graph, every two vertexes of which are backbone connected; a clique graph was defined as a directed graph of maximal joint graph narrowings on weights of vertexes; then a sinew was defined as a graph built over connected components of a minimal joint graph narrowing on a weight of a vertex; a bunch was defined as the union of sinews.
The proposed system of terms let to structure and describe the research space and to find the similarities and the difference between minimal joint graphs that were built over different maximal knowledge pattern sets.
The facts such as a significant narrowing of a minimal joint graph is connected; the weight of edges of a sinew of a significant narrowing is equal to the weight of the narrowing; a set of edges with the same weight is a sinew in a minimal joint graph narrowing of the same weight; all the sinews have the same number of edges; if any significant narrowing of a graph is backbone connected than the graph is the joint graph were proven.
As a result the structure theorem about coincidence of the minimal joint graph set and the bunch set that could be represented as a Cartesian product of sets of subgraphs was proven. This simplifies both the representation and consequential synthesis of all structure elements. The scheme of minimal joint graph set building algorithm and the scheme of improved minimal joint graph set building algorithm was given on the base of the said theorem. Also was proven, that the cardinality of the set is equal to a product of cardinalities of set of sinews, chosen one for every clique and the numbers of edges in every minimal joint graph are similar to each other.
In completion as a result two schemes of minimal joint graph building algorithms are given that consequently synthesize the clique graph, sinew sets for each clique and unite these sinews into a bunch set, thus producing a minimal joint graph set.
The theoretical results give a base for a correct global machine learning (structure synthesis) algorithmization of algebraic Bayesian networks. Particularly, the scheme of the algorithm letting to synthesize a minimal joint graph set for given maximal knowledge pattern set is given.