Bob Thrun wrote:Graph theory is one of the simplest branches of mathematics with the most abstract notation.
So simple, I've never heard of it ;-)
Bob Thrun wrote:A graph consists of edges and vertices (or arcs and nodes, or lines and junctions; the terms are equivalent).
Does this graph have 2 or 3 axes?
Bob Thrun wrote:A cave map is a graph that shows passages (edges) and junctions (vertices). The edges may be stretched, moved, and curved.
Wouldn't the stretching, moving, and curving of the edges change the length of the edges? Wouldn't that be manipulating data to fit a model?
And what about flank margin caves? How would the author define large rooms with pillars as edges and vertices? Wouldn't that be a function of the quality of the data he is looking at, and the survey standard that the survey crew followed? And did the cartographer utilize all the data the survey team collected, or did the cartographer generalize some the the data?
Bob Thrun wrote:If the graph can be flattened so that it lays in a plane without any edges crossing, then the graph is planar.
That would define some caves, but clearly not all. Passages at different elevations can intersect and briefly connect- and much of the definition of how the passages connect and intersect depends on how motivated the surveyors are.
Bob Thrun wrote:An easily described example of a nonplanar graph consists of five vertices, each vertex being connect to all the other vertices.
If you are plotting the five vertices in 2 dimensions, wouldn't that define a splay shot of a room- albeit with a couple of extra shots?
Still grappling with the original quote: ""As far as I know, no cave cartographer has ever shown that any natural cave is nonplanar." I'm pretty sure that's because no one has ever bothered- otherwise there'd be one published paper on the subject somewhere! Someone please send me a link...
Every definition of nonplanar I've found today(including the mathematical ones) would apply to many surveys and maps of many caves. If by 'cave map' the author means a plan view lacking any elevation data- sure, that's planar- and, without parsing through the survey data, and the quality of the survey itself, a potentially inaccurate representation of a three-dimensional space.