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A little bit off-topic but I have a real problem: I sometimes play with my 4 year old son with the Lego Duplo train and I realized that often our tracks have f

A little bit off-topic but I have a real problem: I sometimes play with my 4 year old son with the Lego Duplo train and I realized that often our tracks have fixed points in the sense that if the train starts at a random track piece and random direction, it eventually arrives at a piece where it cannot - traverse the same piece the other way round - reach all other pieces from the same piece The track consists only of standard 2-ended pieces and junctions with 3 ends ABC where AB or AC is an edge but not BC. In order to come up with a track where the above problems do not arise, I'd like to - formalize the track layout and find a suitable logic (which is not that easy given that AB and AC are edges and therefore BA and CA are edges but it does not follow that: BA & AC => BC, i.e., it's not transitive) - find track layouts where each piece is reachable in either direction from every other piece/direction combination on the track. (I know that an 8 where the middle point consists of 2 connected junctions is such a track) Up to now I have been unable to find anything formal and useful other than trivalent graphs [1] (a dead end) and penrose railway mazes [2] where the problem is just solved on images using search algorithms. [1] https://en.wikipedia.org/wiki/Cubic_graph https://en.wikipedia.org/wiki/Cubic_graph [2] https://demonstrations.wolfram.com/PenrosesRailwayMazes/ https://demonstrations.wolfram.com/PenrosesRailwayMazes/ Can anyone help me?

ratmice 8y ago
i'm not sure i understand, but perhaps https://en.wikipedia.org/wiki/Pancyclic_graph https://en.wikipedia.org/wiki/Pancyclic_graph
harrisongsmith 8y ago
I think you are on the right track with the figure 8 configuration. I think the two loops are what are important for traversing both directions. I would formally define this loop as follows. (Note: that since these edges are directional I am using "->" to denote these edges.) Definition: Junction Loop Consider an arbitrary junction K with vertices l,m,n such that l->m and l->n. This junction K will contain a loop if there exists a path that connects m->n or n->m. ------------------------------ Defintion: A Dmichulke Track A Dmichulke Track is a track that any piece of track can be traversed in both directions and there exists a path between any two pieces of track ------------------------------- Conjecture: Any track will be a Dmichulke track if it has a two junctions J_1 (with vertices a,b,c such that a->b, a->c) and J_2 ( with vertices x,y,z where x->y and x->z) that satisfy the following conditions. 1) J_1 and J_2 must be have a path connecting vertex a to vertex x 2) J_1 and J_2 must contain a loop. 3) There must exist a path to either of these junctions from any point of the track. ------------------------------- This should satisfy the stated conditions. You should play around with track configurations that satisfy these conditions and see if you can find a counter example. This is only a conjecture though since I didn't really prove anything. I may take a crack at working on that proof over the weekend and let you know how it goes.

ratmice 8y ago

i'm not sure i understand, but perhaps https://en.wikipedia.org/wiki/Pancyclic_graph https://en.wikipedia.org/wiki/Pancyclic_graph

harrisongsmith 8y ago

I think you are on the right track with the figure 8 configuration. I think the two loops are what are important for traversing both directions. I would formally define this loop as follows. (Note: that since these edges are directional I am using "->" to denote these edges.) Definition: Junction Loop Consider an arbitrary junction K with vertices l,m,n such that l->m and l->n. This junction K will contain a loop if there exists a path that connects m->n or n->m. ------------------------------ Defintion: A Dmichulke Track A Dmichulke Track is a track that any piece of track can be traversed in both directions and there exists a path between any two pieces of track ------------------------------- Conjecture: Any track will be a Dmichulke track if it has a two junctions J_1 (with vertices a,b,c such that a->b, a->c) and J_2 ( with vertices x,y,z where x->y and x->z) that satisfy the following conditions. 1) J_1 and J_2 must be have a path connecting vertex a to vertex x 2) J_1 and J_2 must contain a loop. 3) There must exist a path to either of these junctions from any point of the track. ------------------------------- This should satisfy the stated conditions. You should play around with track configurations that satisfy these conditions and see if you can find a counter example. This is only a conjecture though since I didn't really prove anything. I may take a crack at working on that proof over the weekend and let you know how it goes.