
Algorithms for Planar Graphs and Beyond (2011) - charlysl
http://courses.csail.mit.edu/6.889/fall11/lectures/
======
dmichulke
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?

~~~
harrisongsmith
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.

------
theaeolist
This may sound snarky but it's actually something I would really like to know:
Are there any serious applications of planar graph theory?

~~~
charlysl
From
[https://www.cs.sfu.ca/~ggbaker/zju/math/planar.html](https://www.cs.sfu.ca/~ggbaker/zju/math/planar.html)

 _connecting utilities (electricity, water, natural gas) to houses. If we can
keep from crossing those lines, it will be safer and easier to install.

Connecting components on a circuit board: the connections on a circuit board
cannot cross. If we can connect them without resorting to another layer of
traces, it will be cheaper to produce.

Subway system: if subway lines need to cross, we've got some serious
engineering to do._

Also:

 _The study of two-dimensional images often results in problems related to
planar graphs, as does the solution of many problems on the two-dimensional
surface of our earth. Many natural three-dimensional graphs arise in
scientific and engineering problems. These often come from well-shaped meshes,
which share many properties with planar graphs._

