Fig. 4

Set of maximum matchings associable to a junction: the columns (\({\mathrm{a}}_{\star }\)), (\({\mathrm{b}}_{\star }\)), and (\({\mathrm{c}}_{\star }\)) outline why and how we can associate a set of maximum matchings to any junction through junctions \(j_{1}\), \(j_{2}\), and \(j_{3}\), respectively, from the notional example in Fig. 1. The subfigures actually organize themselves in a table: each column corresponds to a notional junction; each row corresponds to a step of our outline. First step: associate to each street-segment a node—we can place each node at the intersection of the street-segment with a circle centred at the junction so that each graph is a circular graph. Second step: link each node to any node along which it might continue its way (see Fig. 2)—each graph is actually a graph representation of the pairable street-segments. Third step: enumerate all the street layouts which could be achieved as part of a configuration of streets according to the state-of-the-art (see Fig. 3)—the construction of the street layouts follow the scheme used in Fig. 3, while the subgraphs link the so paired street-segment-nodes with a fat red edge. Fourth step: by representing the subgraphs in canonical form, we immediately realize that Step three actually enumerate all the subgraphs with the maximum number of non-adjacent edges, namely, all the maximum matchings—this completes our outline. Notice that the subfigure labels in rows 3 and 4 enumerate the maximum matchings with balanced ternary numbers using down-spin (\(\downarrow\)), nil-spin (0), and up-spin (\(\uparrow\)) as ternary digits (Knuth 1997). This enumeration offers between maximum matchings of junctions joining three or four street-segments and Ising spin states (down \(\downarrow\) and up \(\uparrow\)) a close analogy as junctions \(j_{1}\) and \(j_{2}\) exemplify well here. This analogy appears to hold as well for our less typical junction \(j_{3}\). Figure 5 uses a simpler but more visual enumeration based on the regular polygons—extended with the degenerate regular digon. Each corner (side) represents then a maximum matching