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So I have an un-directed un-weighted graph. It contains cycles. I would like to find the path which visits the most nodes with no repeat visits to any node. Since this is a graph traversal, you can start and end at any node you like.

Background Research: I have looked at Travelling Salesman Problem (TSP); this problem is different and does NOT allow you to finish where you started from and there are no weights. I have looked at several other algorithms, but have found none suitable for this problem.

Graph Size: There are 100 nodes in the graph; with 10 disconnected nodes.

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Maybe you are interested in Returning Paths on Cubic Graphs Without Backtracking... –  draks ... Nov 23 '12 at 20:49

1 Answer 1

This problem is still difficult, it is known as Hamiltonian path, and NP-complete. It doesn't really make a difference if the path is closed (a circuit) or not. Just ask the HAM-PATH problem for every pair of endpoints of all edges and you could solve HAM-CIRCUIT.

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Thanks! What is HAM-PATH, HAM-CIRCUIT is it just your notation or some library in Matlab? –  Chirayu Shishodiya Nov 23 '12 at 20:15
Also there may not be any Hamiltonian path or cycle in my problem, I just want to visit the max number of nodes. –  Chirayu Shishodiya Nov 23 '12 at 20:20
No its just a short cut for the Hamiltonian Path problem, and the Hamiltonian Circuit problem. –  A.Schulz Nov 23 '12 at 20:20
Yeah, but the maximum could be the number of nodes and thus you can detect a Hamiltonian Path with your algorithm. Just ask if the maximum path visits all nodes. –  A.Schulz Nov 23 '12 at 20:23
I think I will write an algorithm of my own which will find the path which visits most nodes. Since there are 90 nodes, there are 90 * 89 combinations of start-end nodes which is roughly 8100 Hamiltonian Path problems; and I am not even sure that a Hamiltonian Path exists. So it could be an exercise in vain if I proceed with finding a Hamiltonian Path problem. –  Chirayu Shishodiya Nov 24 '12 at 8:58

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