graphs and networks - How to find a longest path, which contains as many vertices

I have this graph:

g=Graph[{1, 2, 3, 4, 5, 6, 7, 8, 9, 
  10}, {SparseArray[Automatic, {10, 10}, 
   0, {1, {{0, 2, 5, 7, 9, 11, 13, 15, 16, 18, 
      20}, {{3}, {4}, {1}, {6}, {8}, {5}, {9}, {2}, {9}, {2}, {6}, 
{3}, {5}, {7}, {8}, {4}, 
           {1}, {8}, {1}, {3}}}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 
     1, 1, 1, 1, 1, 1, 1, 1}}], Null}, {VertexLabels -> {"Name"}}]

I want to find a longest path, which contains as many vertices as this path $7\to8\to4\to9\to1\to3\to5\to2\to6$ found by visual inspection. But how do I find it with Mathematica?

Answer

You can just find all the paths by brute force, and use MaximalBy

allPaths = 
  FindPath[g, #2, #1, Infinity, All] & @@@ 
    Subsets[VertexList[g], {2}] // Apply[Join];
MaximalBy[allPaths, Length@Union@# &]
(* {{10, 1, 3, 9, 8, 4, 2, 6, 5}, {7, 8, 4, 9, 1, 3, 5, 2, 6}} *)

Comments

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