Reduction in model search space by enumerating unique topologies as a function of the states.

A) Blue states are non-open while the open state (root) is colored green. All 36 permutations of three state rooted topologies are depicted. Permutations outlined in black represent the original six possible rooted topologies of three states. Orange shading represents the three isomorphic permutations of the six possible rooted topologies while the three unshaded topologies correspond to the unique rooted topologies as in B. The yellow shaded topologies are the remaining 30 permutations. A reduction from 36 rooted graph permutations to three unique topologies is depicted in C. C) Results of a similar graphical enumeration analysis for rooted topologies with 4+ states. D) Biophysically inspired restriction of the maximum degree in a graph to 4. Applying this restriction to the unique topologies results in a reduction in a model search space as enumerated in the table. After further restricting the maximum cycle length in a graph to size 4 after the degree restrictions, final graph counts are displayed in E as function of the number of states. E) Enumeration summary table of rooted graph permutations, rooted topologies, unique rooted topologies, and biophysical restrictions as a function of the number of states.