Feedback Vertex Sets on Tree Convex Bipartite Graphs

A feedback vertex set in a graph is a subset of vertices, such that the complement of this subset induces a forest. Finding a minimum feedback vertex set (FVS) is \(\cal{NP}\)-complete on bipartite graphs, but tractable on chordal bipartite graphs. A bipartite graph is called tree convex, if a tree is defined on one part of the vertices, such that for every vertex in the other part, the neighborhood of this vertex induces a subtree. First, we show that chordal bipartite graphs form a proper subset of tree convex bipartite graphs. Second, we show that FVS is \(\cal{NP}\)-complete on the tree convex bipartite graphs where the sum of the degrees of vertices whose degree is at least three on the tree is unbounded. Combined with known tractability where this sum is bounded, we show a dichotomy of complexity of FVS on tree convex bipartite graphs.