Representation of deterministic graphs by defining pair of words

Abstract

The paper proposes a representation of deterministic graphs (D-graphs) by sets of words in the alphabet of labels of their vertices. Nowadays graphs are a conceptual tool for analytics, development and testing of various software. Among all graphs, there is a subclass of labeled graphs whose elements have labels from a predefined alphabet. Such graphs are actively used to describe and model computational processes in programming, robotics, model verification and validation, etc. Labeled graphs are an information environment for mobile agents, whose movement along the graph can be represented by sequences of vertex labels - words in the label alphabet. A vertex-labeled graph is said to be D-graph if all vertices in the neighborhood of every its vertex have different labels. For such graphs, in the case when its graph map (i.e., the set of vertices and edges and the labeling function) and the initial vertex from which the agents start their movements are known, there is an unambiguous correspondence between the sequence of labels of the vertices visited by the agent and the trajectory of this agent's movements in the graph. In the case when the map of the studied D-graph is unknown to an external observer, the movements of agents can be organized in such a way that, based on their analysis, the observer receives the desired information about the structure of the graph (for example, the map of the graph, the shortest paths between vertices, comparison of the studied graph with the reference graph). Such an analysis can significantly simplify the linguistic representation of a D-graph - the mapping of a graph to one or more finite sets of words in the alphabet of graph vertex labels, which can be used to reconstruct the graph. In this paper, we propose a representation of deterministic graphs by a defining pair of word sets, the first component of which describes the cycles of the graph, and the second - its leaf vertices. This representation is analogous to the system of defining relations for automata. We proposed the algorithm, that for any pair of sets, builds a D-graph for which this pair is deterministic, or reports that it is impossible to do so is given. The algorithm for constructing a canonical defining pair for a D-graph is also given and numerical estimates of this pair for D-graphs with a known number of vertices and edges are found. Further directions of research on this topic are outlined. The results will allow to use new methods and algorithms for solving problems of analysis of graphs with marked vertices.