Feedback vertex sets on restricted bipartite graphs

A feedback vertex set (FVS) in a graph is a subset of vertices whose complement induces a forest. Finding a minimum FVS is NP-complete on bipartite graphs, but tractable on convex bipartite graphs and 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, its neighborhood induces a subtree. When the tree is a path, a triad or a star, the bipartite graph is called convex bipartite, triad convex bipartite or star convex bipartite, respectively. We show that: (1) FVS is tractable on triad convex bipartite graphs; (2) FVS is NP-complete on star convex bipartite graphs and on tree convex bipartite graphs where the maximum degree of vertices on the tree is at most three.