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I am curious whether the following problems has been studied before, but wasn't able to find any papers about it:

Given a planar graph $G$, and two vertices $s$ and $t$, find an $s$-$t$ path $P$ which minimizes the number of distinct faces of $G$ containing vertices of $P$ on their boundaries.

Does anybody know any references?

Thinh D. Nguyen
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Joe
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    This may be a silly question, but would you need to fix an embedding for this? – G. Bach Dec 20 '13 at 02:14
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    @G.Bach: from an algorithmic point of view, when we say "given a planar graph", we usually mean "given a combinatorial map for a graph", hence the embedding is indeed fixed. – zarathustra Dec 20 '13 at 08:03
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    Do you have an application in mind, or just curiosity? – Joe Dec 20 '13 at 21:13
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    which research databases have you searched? – orezvani May 29 '14 at 03:21
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    @zarathustra: No, if we say "given a planar graph" we mean "given a planar graph". If we assume that this graph is equipped with a combinatorial embedding we say, "given a plane graph". Of course, if the graph is 3-connected then the combinatorial embedding is fixed up to a global reflection. – A.Schulz Dec 19 '14 at 08:11
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    @A.Schulz: please give a reference for this distinction. Anyway, the question is meaningful only when an embedding is given. – zarathustra Dec 19 '14 at 17:11
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    @zarathustra you need only go to Wikipedia for this distinction. – Pål GD Apr 20 '15 at 05:19
  • It might be a silly question, but does the required metric (sum of number of faces) change with different embeddings of the given graph? – codeR Feb 07 '24 at 13:17
  • @codeR It's a good question. But yes, this number may change for a certain path. Let $G = ({,0, 1, 2, 3, 4, 5, 6,}, {,{,0, 1,}, {,1, 2,}, {,1, 3,}, {,1, 4,}, {,2, 5,}, {,3, 5,}, {,4, 5,}, {,5, 6,},})$. Let consider path $P = 0, 1, 2, 5, 6$. If both $0$ and $6$ belong to the same face as $2$, then $P$ touches $2$ of $3$ faces only. Otherwise it touches all $3$ faces. Also both the optimal path and the number of faces in the answer may change as well, but my example is too big for a comment – Smylic Mar 28 '24 at 19:30

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