The structure of a painted cactus, insofar as it presents itself to the visual imagination, can be described as follows.  The overall structure, as given by its underlying graph, falls within the species of graph commonly known as a rooted cactus, to which is added the idea that each of its nodes can be painted with a finite sequence of paints, chosen from a palette given by the parametric set

It is conceivable on purely graph‑theoretic grounds to have a class of cacti which are painted but not rooted, so it may occasionally be necessary, for the sake of precision, to more exactly pinpoint our target species of graphical structure as a painted and rooted cactus (PARC).

A painted cactus, as a rooted graph, has a distinguished node called its root.  By starting from the root and working recursively, the rest of its structure can be described in the following fashion.

Each node of a PARC consists of a graphical point or vertex plus a finite sequence of attachments, described in relative terms as the attachments at or to that node.  An empty sequence of attachments defines the empty node.  Otherwise, each attachment is one of three kinds:  a blank, a paint, or a type of PARC called a lobe.

Each lobe of a PARC consists of a directed graphical cycle plus a finite sequence of appendants, described in relative terms as the appendants of or on that lobe.  Since every lobe comes already attached to a particular node, exactly one vertex of the corresponding cycle is the vertex at that node.  The remaining vertices of the cycle have their definitions filled out according to the appendants of the lobe in question.

An empty sequence of appendants is structurally equivalent to a sequence containing a single empty node as its only appendant.  Either way of looking at it defines a graph‑theoretic structure called a needle or a terminal edge.  Otherwise, each appendant of a lobe is itself an arbitrary PARC.

Resources

• Cactus Language • Mechanics
• Survey of Animated Logical Graphs
• Survey of Theme One Program

cc: Academia.edu • BlueSky • Laws of Form • Ma^thstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science