## The Graph type of a Polyomino

This is a continuation of the post radlams.

A graph is some dots (called nodes or vertices) and lines between some of them (called edges). It’s not the geometry of the graph that matters, just how the vertices are connected up by edges.

Associated to each polyomino is a graph, called the graph-type of the polyomino. You get it by putting a vertex at the center of each square and then adding edges between the vertices of adjacent squares:

It turns out (my friend Sean, a graph theorist, pointed this out to me) that whether or not a polyomino is a radlam depends only on its graph type.

Radlams can be pretty easily classified by their graph type. Basically they come in two infinite families. The open ones:

$\qquad\qquad\vdots$

And the closed ones:

$\qquad\qquad\vdots$

If you notice, there are no radlams of size $p$ for $p=4n+2$ except for $p=6$, in which case we have the single radlam:

Making it unique.

In a subsequent post I will outline a proof of this classification theorem. (by the way, I’m pretty sure that I got all of them in the above lists. If not please let me know).