Bond based Description

Introduction

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In this representation, we're more interested in the connections between sites than the sites themselves. We add bonds instead of sites, but otherwise this is similar to the site-based method of building embeddable graphs. Again, we focus on the square lattice.

Generating new graphs

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In this case, we start with one bond, either horizontal or vertical depending on our canonical labelling scheme. Each bond has six possible additions to it, as shown below:

In a similar fashion to our approach with the site based model, we try adding bonds at each possible site until there are no more spots to add a bond (since bonds can't overlap!) then backtrack and start at a new site. It may be more efficient to only add bonds on three of the sites - the ones facing upwards and to the right. Again we can use a canonical labelling scheme to eliminate duplicate graphs. In this scheme, we keep a list of where on the lattice the bonds are, which is enough to encode which sites are bonded together.