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Лингвистическое представление графов с помеченными вершинами Представлено членом-корреспондентом НАН Украины И.И. Скрыпником В работе вводится лингвистическое представление Д-графов, у которых в окрестности каждой вершины все вершины имеют разные метки, определяющей парой множеств слов, первое из которых описывает циклы графа, а второе-все его висячие вершины. Предложена процедура, которая по заданной паре множеств либо строит соответствующий ей Д-граф, либо показывает, что по этой паре Д-граф построить невозможно. Найдены процедура построения минимальной (канонической) определяющей пары для графа и процедура преобразования произвольной определяющей пары графа к канонической. Полученные результаты являются распространением соответствующих задач теории автоматов на графы с помеченными вершинами и позволяют задействовать новые методы и алгоритмы для решения задач анализа графов с помеченными вершинами.

The aim of this paper is to study the representation of deterministic graphs (D-graphs) by sets of words over the vertex labels alphabet and to find metric properties of this representation. Vertex-labeled graphs are widely used in various computational processes modeling in programming, robotics, model checking, etc. In such models graphs playing the role of an information environment of single or several mobile agents. Walks of agents on a graph determines the sequence of vertices labels or words in the alphabet of labels. A vertex-labeled graph is said to be D-graph if all vertices in the neighborhood of every its vertex have different labels. For D-graphs in case when the graph as a whole and the initial vertex (i.e. the vertex from which the agent started walking) are known there exists the one-to-one correspondence between the sequence of vertices visited by the agent and the trajectory of its walks on the graph. In case when the D-graph is not known as a whole, agent walks on it can be arranged in such way that an observer obtains information about the structure of the graph sufficient to solve the problems of graph recognizing, finding optimal path between vertices, comparison between current graph and etalon graph etc. This paper specifies the representation of D-graphs by the defining pair of sets of words (the first describes cycles of the graph and the second -- all its vertices of degree 1). This representation is an analogue of the system of defining relations for everywhere defined automata. The structure of the so-called canonical defining pair, which is minimal in terms of the number of words, is also considered. An algorithm for building such pair is developed and described in detail. For D-graphs with a given number of vertices and edges, the exact number of words in the first component of its canonical defining pair and the minimum and maximum attainable bounds for the the number of words in the second component of this pair are obtained. This representation allows us to use new methods and algorithms to solve the problems of analyzing vertex-labeled graphs.

Automata walking on graphs are a mathematical formalization of autonomous mobile agents with limited memory operating in discrete environments. Under this model a broad area of studies of the behaviour of automata in labyrinths arose and intensively developing last decades (a labyrinth is an embedded directed graph of special form). Research in this regard received a wide range of applications, for example, in the problems of image analysis and navigation of mobile robots. Automata operating in labyrinths can distinguish directions, that is, they have a compass. This paper deals with the problem of constructing square grid graph vertex labelling thanks to which a finite automaton without a compass can walk on graph in any arbitrary direction. The automaton looking over neighbourhood of the current vertex and may travel to some neighbouring vertex selected by its label. In this paper, we propose a minimal deterministic traversable vertex labelling that satisfies the required property. A labelling is said to be deterministic if all vertices in closed neighbourhood of every vertex have different labels. In previous works we have proved that minimal deterministic traversable vertex labelling of square grid graph uses labels of five different types. In this paper we prove that a collective of one automaton and three pebbles can construct this labelling on initially unlabelled infinite square grid graph. We consider pebbles as automata of the simplest form, whose positions are completely determined by the remaining automata of the collective.

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