# 2-connected planar graph

Let $G$ a simple $2$-connected planar graph so that all vertices are incident with the infinite region. Suppose that every bounded region of $G$ has length $3$ (so is a cycle of length $3$). Let $k$ be the number of vertices of degree $2$ in $G$, and let $r$ be the number of regions of $G$ sharing no edges with the infinite region.

If $|V(G)| > 3$, show that \begin{align} k = r + 2 \end{align} I'm trying to figure out how to go about this. So, I think that these could be of help:

• $|E|−|V|=|R|−2$
• Every edge bounds two regions
• If the region shares no edge with the infinite region, then it only shares edges with other regions of length 3.
• $\sum\deg(v)=2|E(G)|$
• $12≤ \sum[6−\deg(v)]$ so $\sum\deg(v)≤6|V(G)|−12$
• for $2$-connected graphs, every vertex has $\deg(v)≥2$

Any ideas?

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• $k=$ degree-1 nodes of the tree,
• $r=$ degree-3 nodes of the tree.
Contract all the degree one tree nodes. This gives almost a full binary tree, if you select one of the degree-3 nodes as root. "Almost", since the root has 3 children. For such a tree we have by induction #leaves=#interior nodes +2. And hence $k=r+2$.