Stick-and-ring graphs (I)

Until now the thread on small graph rewrite systems (last post here) was about rewrites on a family of graphs which I call “unoriented stick-and-ring graphs”. The page on small graph rewrite systems contains several formalisms, among them IC2, SH2 and system X are on unoriented stick-and-ring graphs and chemlambda strings is with oriented edges. Emergent algebras and Interaction Combinators are with oriented nodes. Pseudoknots are stick-and-ring graphs with oriented nodes and edges.

In this post I want to make clear what unoriented stick-and-ring graphs are, with the help of some drawings.

Practically an unoriented stick-and-ring graph is a graph with colored nodes, of valence 1, 2 or 3, which admit edges with the ends on the same node. We imagine that the nodes have 1, 2, or 3 ports and any edge between two nodes joins a port of one with a port of another one. Supplementary, we accept loops with no nodes and moreover any 3-valent node has a marked port.

marked-graphs

If we split each 3-valent node into two half-nodes, one of them with the one marked port, the other with the remaing two ports, then we are left with a collection of disjoint connected graphs made of 1-valent or 2-valent nodes.

marked-graphs-1

These graphs can be either sticks, i.e. they have 2 ends which are 1-valent nodes, or they can be rings, i.e. they are made entirely of 2-valent nodes.

marked-graphs-2

It follows that we can recover our initial graph by gluing along  the sticks ends on other sticks or rings. We use dotted lines for gluing in the next figure.

marked-graphs-4

A drawing of an unoriented stick-and-ring graph is an embedding of the graph in the plane. Only the combinatorial information matters. Here is another depiction of the same graph.marked-graphs-3

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