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 of a problem that does not behave computationally the same in all subclasses of chordal graphs is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph $$G=(V,E)$$ G = ( V , E ) and a set $$S\subseteq V$$ 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 the leafage, which 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 $$\ell $$ ℓ , we provide an algorithm for solving SFVS with running time $$n^{O(\ell )}$$ n O ( ℓ ) , thus improving upon $$n^{O(\ell ^2)}$$ n O ( ℓ 2 ) , which is the running time of an approach that utilizes the previously known algorithm for graphs with bounded mim-width. We complement our result by showing that SFVS is w[1]-hard parameterized by $$\ell $$ ℓ . Pushing further our positive result, it is natural to also consider 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 solving SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs with mim-width one, which is faster than the approach of constructing a graph decomposition of mim-width one and applying the previously known algorithm for graphs with bounded mim-width.

Parameterized algorithms for Steiner tree and (connected) dominating set on path graphs

“…In this article, given the hardness of Dominating Set parameterized by the size of the solution on chordal graphs, we investigate DOMINATING SET and the related problems, CONNECTED DOMINATING SET and STEINER TREE, constrained to the subclasses of chordal graphs known as path graphs. Two structural parameters have been studied for understanding intractable problems on chordal graphs that admit polynomial-time algorithms on the proper subclass of interval graphs: the leafage measures how close a chordal graph is to being an interval graph and the vertex leafage measures the closeness to undirected path graphs [3,28,31].…”