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A feedback vertex set in an undirected graph is a subset of vertices removal of which leaves a graph with no cycles. Razgon (in: Proceedings of the 10th Scandinavian workshop on algorithm theory (SWAT 2006), pp. 160---171, 2006) gave a $$1.8899^n n^{O(1)}$$1.8899nnO(1)-time algorithm for finding a minimum feedback vertex set in an $$n$$n-vertex undirected graph, which is the first exact algorithm for the problem that breaks the trivial barrier of $$2^n$$2n. Later, Fomin et al. (Algorithmica 52:293---307, 2008) improved the result to $$1.7548^n n^{O(1)}$$1.7548nnO(1). In this paper, we further improve the result to $$1.7266^n n^{O(1)}$$1.7266nnO(1). Our algorithm is analyzed by the measure-and-conquer method. We get the improvement by designing new reductions based on biconnectivity of instances and introducing a new measure scheme on the structure of reduced graphs.

In the map verification problem, a robot is given a (possibly incorrect) map M of the world G with its position and orientation indicated on the map. The task is to find out whether this map, for the given robot position and its orientation in the map, is correct for the world G. We consider the world model of a graph G = (V G, E G) in which, for each vertex, edges incident to the vertex are ordered cyclically around that vertex. (This also holds for the map M = (V M, E M.) The robot can traverse edges and enumerate edges incident on the current vertex, but it cannot distinguish vertices (and edges) from each other. To solve the verification problem, the robot uses a portable edge marker, that it can put down at an edge of the graph world G and pick up later as needed. The robot can recognize the edge marker when it encounters it in the world G. By reducing the verification problem to an exploration problem, verification can be completed in O(|V G| × |E G|) edge traversals (the mechanical cost) with the help of a single vertex marker which can be dropped and picked up at vertices of the graph world (G. Dudek, M. Jenkin, E. Milios, and D. Wilkes, IEEE Trans. on Robotics and Automation, vol. 7, pp. 859–865, 1991; Robotics and Autonomous Systems, vol. 22(2), pp. 159–178, 1997). In this paper, we show a strategy that verifies a map in O(|V M|) edge traversals only, using a single edge marker, when M is a plane embedded graph, even though G may not be planar (e.g., G may contain overpasses, tunnels, etc.).