A polynomial time algorithm to compute the connected treewidth of a series–parallel graph - Mescoff, Guillaume; Paul, Christophe; Thilikos, Dimitrios M. (Discrete Applied Mathematics, 2022)
It is well known that the treewidth of a graph G corresponds to the node search number where a team of searchers is pursuing a fugitive that is lazy and invisible (or alternatively is agile and visible) and has the ability to move with infinite speed via unguarded paths. Recently, monotone and connected node search strategies have been considered. A search strategy is monotone if it prevents the fugitive from pervading again areas from where he had been expelled and is connected if, at each step, the set of vertices that is or has been occupied by the searchers induces a connected subgraph of G. It has been shown that the corresponding connected and monotone search number of a graph G can be expressed as the connected treewidth, denoted by ctw(G), that is defined as the minimum width of a rooted tree-decomposition (X,T,r), where the union of the bags corresponding to the nodes of a path of T containing the root r is connected in G. In this paper, we initiate the algorithmic study of connected treewidth. We design a O(n2⋅logn)-time dynamic programming algorithm to compute the connected treewidth of biconnected series–parallel graphs. At the price of an extra n factor in the running time, our algorithm generalizes to graphs of treewidth at most two.