Math: solving outcomes...

Excuse me for being a little off topic as my problem has to do more with math than with programming (though I wouldn't mind if someone would make me an application from this). The one thing that this has to do with programming is that in this one needs to describe a problem (isn't that what programming is all about?) and I fail to do so - analytically. But here's what I got (to get to the point): I need to solve how many different graphs I get when

a) The grid where the graph is drawn is a hexagon with a vertical line splitting and two horizontal lines going through, so it looks like this

Code:

` /|\ Here two (back)slashes represent one`

/_|_\ and two overlapping '|' equal one

|_|_| (needed to get those undescores to fit)

\ | /

\|/

So there is a 13 possible lines that can be drawn to the grid

b) The graph is drawn from a single line (so every line drawn in to the grid is connected to another)

c) There can be no branches in the graph (and therefore the line can't cross itself)

d) The graph must go to every four part of the grid (E,W,N,S) at least once

e) (And to make it easier - and still just enough for my needs) the graph must be drawn so that no lines can be added to it without breaking the rule c

x) And if it's not too much trouble no graph can't be the mirror and/or flipped image of another graph

Here's a few example to (hopefully) make it a bit clearer (hope this fits to your screeen):

Code:

` /\ The simplest /\ Wrong, /|\ Wrong, \ Little bit / Wrong,`

/ \ outcome / \ south / | \ branches ___\ advanced / line is

| | |__| is not | | | at north | graph | not

\ / reached \ | / and south \ / / coherent

\/ \|/ \/ /

I've done some programming myself, a simple sokoban-game, game of life and that sort of stuff but I find this problem to be just a bit too much for my skills. So, what I'm asking for you to do is... Well I actually don't know what I'm really asking - the best you could do is to solve every graph according to my rules (my guess is that the number of these is a bit over twenty). But I'd also appreciate if you could tell me the exact number of all possibilities. If you do know how to solve it, but are lazy enough to not do it, please tell me and I'll do the work. Perhaps a simple snippet that produces these graphs would do so verifying the x condition would be a hell of a lot easier than it would be by twiddling just numbers.

If you've have gotten this far I thank you for your time and wish you good luck if you are to try solving this. Cheers!