# 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 ctw(G), that is defined as the minimum width of a rooted tree-decomposition ( 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 tw(G)≤ctw(G). It is paper, we initiate the algorithmic study of connected treewidth. We design a O(n^2·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.

READ FULL TEXT