first, consider roads drawn on 1x1 square tiles (monominoes). the midpoint of each edge is a connection point for a road going from one tile to an adjacent tile. examples: Truchet tiles, Carcassonne. label the sides of the square 0 1 2 3 in clockwise order. there are 7 possible types of tiles, not counting rotations:

p (0123)

p (01 23)

(02 13)

p (0 123)

p (0 1 23)

p (0 2 13)

p (0 1 2 3)

each cluster of digits represents a connected component of the graph drawn on the tile. a singleton connected component means there is no road along that side. (we'll consider another possibility later.)

the prefix "p" indicates the graph is planar. the only non-planar tile is the one with two roads going straight across at right angles, one flying over the other.

next, consider domino tiles. 6 possible connection points for roads along the sides of the domino. label them 0 through 5 around the perimeter. 0 and 3 are the short sides.

omit duplicates from 180 degree rotation.

omit from enumeration domino tiles whose road network can be represented by two monomino tiles. determine this as follows. consider the line segment dividing a domino into two squares. keep a domino if the road graph crosses the middle line segment at least twice. a connected component contributes one crossing if and only if it has nodes on both squares. for two crossings, there needs to be two connected components, both which cross the middle edge.

p (012 345) & (045 123)

(013 245) & (035 124)

(014 235)

(02 1345) & (04 1235)

(024 135)

(03 1245)

p (12 0345)

(14 0235) & (25 0134)

(02 13 45)

(02 14 35) & (04 13 25)

p (03 12 45)

(03 14 25)

p (0 12 345) & (0 45 123)

(0 13 245) & (0 35 124)

(0 14 235) & (0 25 134)

p (1 02 345) & (5 04 123)

(1 03 245) & (5 03 124)

(1 04 235) & (5 02 134)

(1 25 034) & (5 14 023)

(1 35 024) & (5 13 024)

p (1 45 023) & (5 12 034)

p (0 2 13 45) & (0 4 12 35)

(0 2 14 35) & (0 4 13 25)

p (0 3 12 45)

(0 3 14 25)

p (1 2 03 45)

(1 2 04 35)

p (1 4 02 35) & (2 5 04 13)

(1 4 03 25) & (2 5 03 14)

there are 47 possible domino tiles, counting chiral pairs as separate. chiral pairs are given on the same line separated by ampersand. (there are 29 if reflections are omitted.) 16 out of 47 are planar, marked "p". (if a tile is planar, so is its mirror image.)

future work: enumerate 2x2 big squares, omitting those which can be broken down into combinations of 1x1 or 1x2. perhaps also omit 3 square L, enumerating them first.

previously, we had declared that a singleton connected component represents no road through that side. alternatively, it could be a short stub of a dead-end road. allowing both possibilities, there are the 18 possible 1x1 square tiles:

p (0123)

p (01 23)

(02 13)

p (0 123)

p (123)

p (0 1 23)

p (0 23) & (1 23)

p (23)

p (0 2 13)

p (0 13)

p (13)

p (0 1 2 3)

p (0 1 2)

p (0 1)

p (0 2)

p (0)

p ()

the presence of a singleton connected component indicates a stub road; the absence of a side being mentioned at all means no road goes through that side.

future work: enumerate the domino tiles permitting both stub roads and no road. do we permit a short purely internal stub road across the middle of the tile? (such a road could be constructed from two monominoes.) similarly, do we permit a purely internal circuit inside a 2x2 tile?

much simpler: the only 1x1 tile is the 4-way intersection (or roundabout). similarly, the domino tile is a 6-way elliptic rotary. the 2x2 tile is a big 8-way rotary, maybe with decoration in the center. also include 3-square L? everything connects to everything.

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