This is my home work problem but i dont have any clue how to proceed with this A “geometric graph” is a special type of graph where the nodes are points on a 2- dimensional surface and edges are straight lines joining pairs of nodes. Show that the minimum spanning tree of such graphs cannot have edges that cross each other (other than at their endpoints).
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You can prove that using the fact that the distance between A and B is always less than or equal to the distance of A and C plus the distance of C and B. If there were edges crossing in the MST, that would contradict this rule. – Ulrich Eckhardt Oct 10 '15 at 15:14
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What does it mean straight line on a surface? Do you mean on a 2D plane? What is straight line on sphere or on hyperbolic paraboloid?
It does not hold in general. Example is graph that is a tree which edges cross. It's minimum spanning tree is same graph. E.g four nodes on a plane like
O-----O
\ /
\ /
X
/ \
/ \
O O
Ante
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That tree is not a *minimum* spanning tree in my definition, but maybe that just comes from the assumption that I have a complete graph (all nodes are connected to all other nodes) to start with. – Ulrich Eckhardt Oct 10 '15 at 15:17
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I have this answer from my algorithm's TA for the same question, let me know if it helps:
The idea is that, if a path contains two edges that cross each other, we can replace those crossing edge with some other edge to get a smaller path.
For example, if there are two edges ac and bd that cross each other, we can replace them with edge ab and cd and get smaller path length.
In geometric graph there is an edge between every pair of vertexes, they also follow triangle inequality.
Pawan
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