Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage

Abstract

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 vertex-weighted graph G = (V,E) and a set S ⊆ V , we seek for a vertex set of minimum weight that intersects all cycles containing a vertex of S. SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains NP-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 nO(ℓ), thus improving upon nO(ℓ2), the running time of the previously known algorithm obtained for graphs with bounded mim-width. We complement our result by showing that SFVS is W[1]-hard parameterized by ℓ. Pushing further our positive result, it is natural to consider a slight generalization of leafage, the vertex leafage, which measures the minimum upper bound on the number of leaves of every subtree in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains NP-complete on undirected path graphs, i.e., chordal graphs having vertex leafage at most two. Lastly, we provide a polynomial-time algorithm for SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs of mim-width one, which is faster than the previously known algorithm obtained for graphs with bounded mim-width.