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 n O(ℓ) , thus improving upon n O(ℓ 2 ) , 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.

Domination and Cut Problems on Chordal Graphs with Bounded Leafage

“…Fomin et al [19] and Arvind et al [2] proved, respectively, that the Dominating Set and Graph Isomorphism problems on chordal graphs are FPT parameterized by the leafage. Barnetson et al [4] and Papadopoulos and Tzimas [44] presented XP-algorithms running in time n O(ℓ) for Fire Break and Subset FVS on chordal graphs, respectively. Papadopoulos and Tzimas [44] also proved that Subset FVS is W [1]hard when parameterized by the leafage.…”

“…Barnetson et al [4] and Papadopoulos and Tzimas [44] presented XP-algorithms running in time n O(ℓ) for Fire Break and Subset FVS on chordal graphs, respectively. Papadopoulos and Tzimas [44] also proved that Subset FVS is W [1]hard when parameterized by the leafage. Hochstättler et al [30] showed that we can compute the neighborhood polynomial of a chordal graph in n O(ℓ) -time.…”