# Questions tagged [planar-graphs]

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### Partitioning the vertices of a graph into induced trees

I am looking for previous work regarding graphs whose vertices can be assigned colours (not necessarily a proper colouring) in such a way that each colour class induces a tree.
In particular I am ...

**3**

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### Reference on generalization of plane graph duality between bonds and simple cycles

Let $G$ be a plane graph, and $G^*$ its dual. Among the $k$ partitions of the nodes of $G$, I'll call the connected k-partitions those such that each block of nodes of the partition induces a ...

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### Reconstructing plane graphs from degree- and face-sequences

Let $G$ be a plane $3$-connected graph; so it partitions the plane
into regions bounded by faces.
Let $\mathrm{deg}_v$ be the sequence of vertex degrees of $G$,
and $\mathrm{deg}_f$ be the sequence of ...

**2**

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34 views

### Volume interpretation of number of perfect matchings in bipartite planar graphs

Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...

**1**

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107 views

### Smallest planar graph with two non-homoemorphic plane embeddings?

Apologies for asking a question which probably has a well-known answer:
What is the smallest (not necessarily simple) planar graph with two non-homeomorphic embeddings into the plane?
I am ...