Densest k subgraph problem for outerplanar graphs? The densest k subgraph problem aims to find a subgraph $H$ of a graph $G$ with exactly $k$ vertices that maximizes the number of edges $|E(H)|$.

Does anyone know if there exists a polynomial-time algorithm for this problem under the restriction that $G$ is outerplanar? (Note: I am specifically asking for an algorithm, not a PTAS, and I want $G$ to be outerplanar, not $b$-outerplanar for some $b &gt; 1$).
It can be solved in linear time in an even more general class of graphs: As shown in

N. Bourgeois, A. Giannakos, G. Lucarelli, I. Milis, V.T. Paschos
Exact and approximation algorithms for densest $k$-subgraph
WALCOM’13, LNCS, vol. 7748, Springer-Verlag (2013), pp. 114-125

the Densest-$k$-Subgraph problem can be solved in $O(2^{\mathrm{tw}(G)}\cdot k \cdot ((\mathrm{tw}(G)^2)+k)\cdot |X|)$ time when a tree decomposition $(X,T)$ of the input graph $G$ is given. Since outerplanar graphs have treewidth at most two, the Densest-$k$-Subgraph can be solved in $O(k^2 n)$ time on outerplanar graphs with $n$ vertices.