A polynomial time algorithm to compute the connected tree-width of a series-parallel graph

It is well known that the treewidth of a graph $G$ corresponds to the node search number where a team of cops is pursuing a robber that is lazy, visible and has the ability to move at infinite speed via unguarded path. In recent papers, connected node search strategies have been considered. A search stratregy is connected if at each step the set of vertices that is or has been occupied by the team of cops, induced a connected subgraph of $G$. It has been shown that the connected search number of a graph $G$ can be expressed as the connected treewidth, denoted $\mathbf{ctw}(G),$ that is defined as the minimum width of a rooted tree-decomposition $({{\cal X},T,r})$ such that the union of the bags corresponding to the nodes of a path of $T$ containing the root $r$ is connected. Clearly we have that $\mathbf{tw}(G)\leqslant \mathbf{ctw}(G)$. It is paper, we initiate the algorithmic study of connected treewidth. We design a $O(n^2\cdot\log n)$-time dynamic programming algorithm to compute the connected treewidth of a biconnected series-parallel graphs. At the price of an extra $n$ factor in the running time, our algorithm genralizes to graphs of treewidth at most $2$.