A polynomial time algorithm to compute the connected tree-width of a series-parallel graph
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by
Guillaume Mescoff, Christophe Paul, Dimitrios Thilikos
2021
Abstract
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 ๐๐ญ๐ฐ(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 ๐ญ๐ฐ(G)โฉฝ๐๐ญ๐ฐ(G). It
is paper, we initiate the algorithmic study of connected treewidth. We design a
O(n^2ยทlog 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.
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