each puzzle consists of a set of images (separated by dotted lines), presented in no particular order, except roughly by the visual complexity, and not necessarily having anything in common other than that they were convenient to construct (tho each puzzle might have some common property, depending on requests, and this will be noted). each image should be considered a single directed graph.
requests may be made for more data points, "natural" subsets, etc. where naturality is about descriptions and relationships between them.
there is no goal other than to understand.
puzzle 2 (natural subset of puzzle 1)
q: obviously some category theory nonsense
a: nope
q: they are a random subset of a set of graphs?
a: no
there is some (possibly infinite) set of graphs S (for each puzzle) of which the graph depicted in each image in the puzzle is a member. this set is describable in an elegant fashion, and puzzles are related to one another also in an elegant fashion, related to these descriptions.
q: are there any subgraphs of those given which are in the set but not themselves given?
a: no
q: do all graphs in the set have 4n vertices and an even number of edges?
a: 4n vertices, yes. even number of edges, not sure.
q: are all the graphs subgraphs of a square grid?
a: yes
q: every vertices should be a part of exactly one even cycle?
a: yes
q: is every component always a 2n-cycle?
a: yes
q: can there be larger than 4-cycle components?
a: yes
q: is the graph consisting of just two 6-cycles a member of the set?
a: no
q: is the graph containing just 4 2n-cycles always a member?
a: no
q: do you have a counterexample?
a: 4 6-cycles
q: is there any element of the set consisting solely of 6-cycles (any number)?
a: yes
q: (preferrably smallest) example?
a: 6 6-cycles
q: what about 2n 2n-cycles in general?
a: yes. this constitutes a natural simplification. mind you, not an atomic one.
q: is the disjoint union of two examples always an example?
a: yes
q: is there an example containing 6-cycles, but less than 6 of them?
a: yes
q: is the graph consisting of just 3 4-cycles and 2 2-cycles a member?
a: yes
q: does this set stem from a problem that is not immediately graph-related?
a: sort of. the way im conceptualizing it is only tangentially graph related.
q: is it a matching problem of sorts?
a: not as im conceiving it but equivalent to one, i have no idea.
q: is it related to the continued fraction? or number theory?
a: doubtful.
q: is it (based on) a counting problem?
a: you probably can think of it as one, but that's probably one of the least intuitive ways to understand what's going on.
q: transportation/travelling salesman stuff?
a: no
q: can you always add a 4-cycle to a graph and still have a member?
a: no idea.
q: is the 1 2-cycle + 1 6-cycle is a member?
a: no.
if a graph is displayed in a puzzle, and some subgraph is not displayed, it means that the subgraph in question is not a member.
for the most part, if a natural subset of a puzzle is displayed, everything in the superset that is currently visible and that is part of the subset will also be visible in the subset, and conversely (i.e. i won't mislead about what is and isn't knowable across puzzles).
q: if you replace a 6-cycle by a 2-cycle, is that still a member?
a: maybe. i'm not sure.'
q: is the membership related to whether the graph can be embedded in some larger structure?
a: it's possible, but i have no idea.
q: is it related to linguistics?
a: no.
q: related to anything other than math?
a: depends on what you eman by math.
q: something non-abstract?
a: yes, i suppose.
q: obviously there are no sinks or sources, but is it related to something flowing between the vertices?
a: ehhhh sort of.
think of what graphs are, beyond merely graphs