I am pursuing generalizations of planar Euclidean geometry to complete symmetric and weighted graphs, the guiding principle being applicability to the TSP. The operations and tests that are available in that generalized geometry are
- adjacency of vertices and edges
- relative orders of edgeweight sums that are independent of vertex-weights.
So far I managed to generalize the notion of the diagonals of the convex hull to those edges that can't be contained in any 2-optimal tour.
Further progress was also possible in understanding the meaning of the Euclidean convex hull when viewed from its generalization and from that formulating conjectures about extensions of the euclidean convex hull.
But the whole subject is still quite nebulous and it is not clear what thinking about the TSP in geometric terms will leverage, therefore my humble questions is about an appropriate name for a "geometry" on graphs as described. I looked up the greek word for network and it seems fit for the subject, but I don't want to redefine a possibly already existing name.
Questions:
- is there already an established name for a geometry on complete symmetric and weighted graphs whose definitions are compliant with the TSP, i.e. are independent of vertex-weights and are not based on embeddings into metric spaced via the construction of vertex coordinates.
- would díktyometry from Greek δίκτυο for network be appropriate? I'm hesitant to insert the term "geo" before "metry" because the meaurements are solely in the network and there is no earth to measure on.
