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Given a multigraph $G$ embedded in the plane call a maximal set of parallel edges between $u,v$ such that only one of the induced faces (a face is a connected region of $\mathbb{R}^2 \backslash G $) contains nodes besides $u$ or $v$ a topologically parallel set (tell me if there is standard terminology for this).

Given a topologically parallel set $S$ of edges between $u$ and $v$ we say that an edge $e$ dipsinto the set $S$ if $e$ intersects some but not all edges of $S$. transformation of <span class=$\phi $ into $\phi'$" />

Is it true that Given a multigraph $G$ with an embedding $\phi$, there is an embedding $\phi'$ with $\phi(V) = \phi'(V)$, preserving the topologically parallel sets such that no edge $e$ dips into a topologically parallel set. Further if two edges cross in $\phi'$, then they cross in $\phi$.

I'm fairly sure this is true simply perturb the drawing so that edges no longer dip into topologcially parallel sets.

Hao S
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  • As I see, the edges of the embedded graph can intersect (and then I would rather speak about a drawn graph instead of the embedded one). But then it is not clear what is a face of the graph. – Alex Ravsky Jul 30 '24 at 01:35
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    @AlexRavsky a face is a a connected region of $\mathbb{R} \backslash G $ the plane with the graph removed. When I say a face contains a node I mean the closure of a face contains a node in it's interior. – Hao S Jul 30 '24 at 02:27
  • There is a standard terminology: Homotopic parallel edges are defined to not have any vertices in the bounded components they leave after removing the closed curve obtained by concatenating them. If you are on the sphere instead of the plane you would say that one component contains all vertices. So what you describe are pairwise homotopic parallel edges. – F.U.A.S. Jul 31 '24 at 15:49
  • @F.U.A.S. Is there also some standard literature talking about this? – Hao S Jul 31 '24 at 16:29
  • @Hao S. There is a few notable papers of Janos Pach. The first one is "Crossings between non-homotopic edges". The second one is "The number of crossings in multigraphs with no empty lens." In this paper the authors explicitly point out that there is a certain difference between the two notions, because homotopic comes from homotopy theory and means, that the two edges cannot be continuously deformed into each other. If you allow for self-intersecting edges your notion is actually different from this notion. – F.U.A.S. Aug 02 '24 at 13:31
  • @F.U.A.S. I saw that but didn't see them talk about reducing to the non-homotopic case which made me slightly suspicious at first as it seems like an important (although possibly simple) selling point of their paper and point to acknowledge in general. Also you can get rid of self intersections by rerouting edges. So all graphs have a nonhomotopic simplification. – Hao S Aug 03 '24 at 01:29

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