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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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1 Answers1

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First off, note that this problem is only well-specified for a given planar embedding (or plane graph) of the planar graph, the planar graph itself is not enough as it doesn't encode which vertices share which incident faces.

Suppose we know and can access the incident faces of any given vertex of the graph in the given embedding, for example by a map $f:V\rightarrow F$ or a bipartite graph or similar suitable data structure.

The problem can then be solved with a slight modification of Dijkstra's algorithm: For every vertex $v$, we have to store a collection of path face sets that are incurred by some $s$-$v$ path. We expand nodes in priority by minimum cardinality of their attached face sets, propagating to the neighbours of the current vertex by taking the set union of each path face set with that neighbor's vertex face set. Terminate once some path face set at vertex $t$ is up for expansion.

For convenient solution extraction, one might want to store additional information in the data structure. E.g. instead of a mere collection of path face sets, store a map from path face sets to the neighbors of the vertex such that each path face set points at a neighbor from which it originated, or even a full path up to that point.

There's some optimizations possible, like keeping only the Pareto optimal path face sets to conserve memory, or doing lazy computation on the expansion step, but it should always return an optimal solution.

MarioVX
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