    # Computing Subset Feedback Vertex Set via Leafage

Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example that behaves computationally different in subclasses of chordal graph is the Subset Feedback Vertex Set (SFVS) problem: given a graph G=(V,E) and a set S⊆ V, SFVS asks for a minimum set of vertices that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains -complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider variants of the leafage that measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage ℓ, we provide an algorithm for SFVS with running time n^O(ℓ). Pushing further our positive result, it is natural to consider a slight generalization of leafage, the vertex leafage, which measures the smallest number among the maximum number of leaves of all subtrees in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains -complete on undirected path graphs, i.e., graphs having vertex leafage at most two. Moreover, we strengthen previously-known polynomial-time algorithm for SFVS on directed path graphs that form a proper subclass of undirected path graphs and graphs of mim-width one.