# Is it possible to define a directed graph “facing” direction?

I'm just trying to learn about graph theory and I'm not strong in math to this degree so I'm wondering if this is even possible. If so, I'd appreciate some beginner level resources, if they exist for this topic. If I have two sets of directed graphs with 2 nodes each (to simplify the example) and they are effectively "pointing" at each other can I determine this in any way? Is there an idea of direction that exists outside of the graph itself? I'm thinking of maybe using simple cardinal directions or just left/right/up/down but since an edge that doesn't exist yet has no direction is this even possible?
I hope this makes some sense and please ask for further clarification since I'd like to refine this idea as I learn more. Thanks!

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It's hard to understand what you mean by "pointing". Could you perhaps give a concrete example of the situation you're trying to describe? – Henning Makholm Dec 11 '11 at 4:53
I'm struggling with an example, but I'm thinking of something like a set of kids train tracks. If I were to lay tracks down in a direction and someone else were laying them down in another direction, is it possible to tell ahead of time that they will connect or if they are just each going off in completely incompatible directions? Maybe "heading" is a better word? – McArthey Dec 11 '11 at 5:07

Graphs don't really work like that. They're just abstract nodes with edges between them. A directed edge points from one node to another. It does not in any way point beyond that node. If you have two nodes, A and B, in a directed graph then an edge which connects to both A and B must point from A to B or from B to A. If there's another node C, the edge doesn't point "towards" or "away from" C. They're not like vectors. It has no relationship to C at all. As such, directed graphs can't point towards each other. It's not a sensible idea in graph theory.

Direction in a directed graph is usually expressed using the metaphor of one-way roads. Let's talk about a small country called Onewayvia ruled by a king (Humperdink IV) who is an adherent of Unidirectionality, a small religious sect who believes that since there is only one way to God, roads should never go both ways. All roads in the kingdom must be one-way roads. Now, it's a small kingdom made of small towns. And, in fact, the towns are so small, that they don't really have any roads inside them, just roads which run between towns and meet in the center of town. And due to a ruling from the Unidirectionality synod of 1242 (the "wacky synod" as it's called in the history books), they've also decided that roads should never ever, ever meet out in the country and should never connect more than two towns together.

Now, let's say given this information, I ask you to make a detailed map. Well, you're going to have a lot of trouble doing that. You don't know what's north or south or east or west of anything else. You don't know latitude or longitude or Cartesian or radial coordinates or distances between the towns or anything else helpful like that. But if I were to ask you a question like "I'm in Ai and I want to get to Dee, how do I get there?" Well, you could answer that question using the information above. You could say "Well, from Ai, you take the road to Bea (the one with all the bridges) and then from Bea, you take the road to Dee (the one with all the tunnels)." So you do actually know how to get around the barony. You can figure out, from that information, how to get from any town to any other.

But to do this, you'd probably want to draw something out because it's tough to keep all that in your head if you haven't driven around the barony (which I don't recommend actually doing). What you'd draw out is a graph-theory style directed graph. On that graph, the towns would be the vertexes (or "vertices" if you want to sound more mathematiciany) and the roads would be the edges. You can draw out a set of dots for the towns and then arrows between them for the roads to show the connections even without knowing the actual positions of the towns or layouts of the roads. That's what we do in graph theory, we map out connections without knowing anything about location or compass direction or anything else like that. The arrangement of the vertexes is completely arbitrary but the connections between them are accurate.