An improved algorithm for the vertex cover P3P_3 problem on graphs of bounded treewidth

Given a graph $G=(V,E)$ and a positive integer $t\geq2$, the task in thevertex cover $P_t$ ($VCP_t$) problem is to find a minimum subset of vertices$F\subseteq V$ such that every path of order $t$ in $G$ contains at least onevertex from $F$. The $VCP_t$ problem is NP-complete for any integer $t\geq2$and has many applications in real world. Recently, the authors presented adynamic programming algorithm running in time $4^p\cdot n^{O(1)}$ for the$VCP_3$ problem on $n$-vertex graphs with treewidth $p$. In this paper, wepropose an improvement of it and improved the time-complexity to 3^p\cdotn^{O(1)}. The connected vertex cover $P_3$ ($CVCP_3$) problem is the connectedvariation of the $VCP_3$ problem where $G[F]$ is required to be connected.Using the Cut\&Count technique, we give a randomized algorithm with runtime$4^p\cdot n^{O(1)}$ for the $CVCP_3$ problem on $n$-vertex graphs withtreewidth $p$.Comment: arXiv admin note: text overlap with arXiv:1103.0534 by other author