The subclass of 2D cellular automata with one graph of connections

We describe how the rules for binary two-dimensional cellular automata with von Neumann neighborhood can be created depending on the appropriate graphs of connections. Those graphs are closely connected with configurations for which the new state of automaton cell stays one. We describe the group of rules created for the chosen symmetric graph which is the same for a few groups of configurations. We analyze how they influence the behavior of cellular automata. We observed that almost all cellular automata with one symmetric graph process the input signal in similar way. Thanks to that, in the analyses of other such subclasses of cellular automata (with other symmetric graphs of connections) we can limit ourselves to the examination of the final state achieved by only one cellular automaton with representing this group rule. It lets us make the process of searching cellular automata with the complex behavior easier.