Supporting Information

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Supporting Appendix
Supporting Figure 3
Supporting Figure 4
Supporting Figure 5




Supporting Figure 3

Fig. 3. Part of a graph where edges (v,w), (v,u), and (u,w) form a tooth. The only nonloop directed into u is the edge (v,u) and the only nonloop directed out of u is the edge (u,w).

Supporting Figure 4

Fig. 4. Part of graph G₀. G₀ is a reflexive graph composed of a directed cycle in which each edge is a part of a tooth. It includes q = 2ⁿ nodes and 5q/2 edges in total. G₀ is used as an element in the construction of G for a pedagogical purpose.

Supporting Figure 5

Fig. 5. With q = 4, G₀ takes the form here; then G is constructed from five differently labeled versions of this form by taking their product and deleting the edge e = [(u₁,…,u₅),(w₁,…,w₅)]. A situation similar to that of "Buridan’s ass" arises when the circuit C is given origin (u₁,…,u₅) and target (w₁,…,w₅). Unable to choose among the allowed two-step paths (where some subcircuits move to their respective w_i’s in the first step and the others in the second step), C attempts to follow the deleted edge e (i.e., all subcircuits attempt to move to their respective w_i’s simultaneously). This move is illegal, and C remains frozen at (u₁,…,u₅).