I am trying to show that every 2-terminal SP-graph is $O(\log(n))$-outer-planar for a challenge question on my assignment. In particular, I am trying to prove this by induction on the number of combinations (either a series or a parallel combination). This is what I have so far.
The base case on $0$ combinations is that we just have some edge $(s, t)$. Clearly this is outer-planar and so $O(\log(n))$-outer-planar as well (in this case, $\log(2) = 1$, so this does hold).
Now we do the inductive step. Given some 2-terminal outer-planar graph $G = (V, E)$, we will show that it is $O(\log(n))$-outer-planar, where $|V| = n$. Since $G$ is a 2-terminal outer-planar graph, it was either the result of combining two graphs $G_1$ on $n_1$ vertices and $G_2$ on $n_2$ vertices by series or by parallel. So we have two cases two consider.
The first case is when $G_1$ and $G_2$ were combined by series. By induction, $G_1$ is $O(\log(n_1))$-outer-planar and $G_2$ is $O(\log(n_2))$-outer-planar. Since $G$ was combined by series, it has $n = n_1 + n_2 - 1$ vertices, where $n_1, n_2 \ge 2$. Let $n_0 = \max(n_1, n_2)$. Then $G$ is $O(\log(n_0))$-outer-planar since we have not added any layers to $G_1$ or $G_2$ by combining them. It follows that $G$ is $O(\log(n))$-outer-planar since $n_0 \le n$.
In the second case, $G_1$ and $G_2$ are combined in parallel, meaning that $G$ has $n = n_1 + n_2 - 2$ vertices. Again by induction, $G_1$ is $O(\log(n_1))$-outer-planar and $G_2$ is $O(\log(n_2))$-outer-planar. However, this time we cannot conclude that the number of layers around $G$ is the same.
This is where I am stuck. I understand that we are at-most doubling the amount of vertices in $G$ by combining them in parallel, but I don't know how to use that to argue that the number of layers in $G$ is $O(\log(n))$. Any help would be appreciated since I've been stuck on this for several days now.
EDIT: Now that the deadline has passed for my assignment, I would love to have a solution so that I can prepare for my midterm exam (solutions to bonus/challenge problems are never posted).
Series-parallel graphs (SP-graphs) are constructed in the following way. A two-terminal graph is a graph with two distinguished nodes $s,t$. A single edge connecting $s$ and $t$ is an SP-graph. If $(G_1,s_1,t_1)$ and $(G_2,s_2,t_2)$ are two SP-graphs, then their serial composition is an SP-graph, and their parallel composition is an SP-graph. The serial composition is obtained by identifying $t_1$ with $s_2$; the new terminals are $s_1$ and $t_2$. The parallel composition is obtained by identifying $s_1$ and $s_2$, and $t_1$ and $t_2$; the new terminals are $s_1=s_2$ and $t_1=t_2$. SP-graphs are the minimal class of graphs generated by these operations. See Wikipedia for more on this class.
$k$-outerplanar graphs: A planar embedding (i.e. an embedding without crossings) is $1$-outerplanar if all of the vertices belong to the unbounded face. For $k \gt 1$, a planar embedding is said to be $k$-outerplanar if removing the vertices on the outer face results in a $(k − 1)$-outerplanar embedding. A graph is $k$-outerplanar if it has a $k$-outerplanar embedding.