A connected feedback vertex set of a graph is a connected subgraph of the graph whose removal makes the graph cycle free. In this paper, we provide an approximation algorithm for connected feedback vertex set in AT-free graphs. Given an \(\alpha \) -approximate solution for feedback vertex set on 2-connected AT-free graph, our algorithm produces a solution of size \(((\alpha +0.9091)OPT+6)\) for connected feedback vertex set on the same graph. The complexity of our algorithm is \(O(f(n)+(m+n))\) , where the time required to obtain the \(\alpha \) -approximate solution is O(f(n)). Our result leads to the following two observations. The optimal feedback vertex set algorithm for AT-free graphs combined with our result provides an algorithm which produces a solution of size \((1.9091OPT+6)\) with running time \(O(n^8m^2)\) for 2-connected AT-free graphs. The 2-approximation algorithm for feedback vertex set in general graphs along with our result provides an algorithm which produces a solution of size \((2.9091OPT+6)\) with running time \(O(min\{m(log(n)),n^2\})\) . Using the same method we also obtain a \(((\alpha +1)OPT+6)\) -approximation for this problem on general AT-free graphs. We note that, the complexity status of this problem is not known.
