Summary: Based on the mobile automaton model, an algorithm is introduced that grows planar, tri-valent graphs by exhibiting a peculiar, twofold dynamics. In a first phase, graph growth appears to be pseudo-random and \(O(n)\) then it settles to a very regular behavior and \(O\sqrt{n}\) rate. A pseudo-random \(O\sqrt{n}\) mobile automaton is already known; the new automaton provides now a finite, but surprisingly long, pseudo-random, linear growth process. Applications of mobile automata to fundamental physics and quantum gravity have been recently suggested.