Linear Time Recognizer for Subsets of ℤ²

Abstract

Graph automata were first introduced by P. Rosenstiehl [5], under the name of intelligent graphs, surely because a network of finite automata is able to know some properties about its own structure. Hence P. Rosenstiehl created some algorithms that find Eulerian paths or Hamiltonian cycles in those graphs, with the condition that every vertex has a fixed degree [6]. Those algorithms are called ”myopic” since each automaton has only the knowledge of the state of its immediate neighborhood. A. Wu and A. Rosenfeld ([7] [8]) developed ideas of P. Rosenstiehl, using a simpler and more general formalism: the d-graphs. Hence, a graph automata is formed from synchronous finite automata exchanging information according to an adjacency graph. A. Wu and A. Rosenfeld gave a linear algorithm allowing a graph automata to know if its graph is a rectangle or not, then, E. Rémila [3] extended this result to other geometrical structures.