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

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Combinatorial Algorithms (IWOCA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13270))

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Abstract

Given a vertex-weighted graph \(G=(V,E)\) and a set \(S\subseteq V\), the Subset Feedback Vertex Set (SFVS) problem asks 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 \(\mathsf {NP}\)-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 \(\ell \), we provide an algorithm for SFVS with running time \(n^{O(\ell )}\), thus improving upon \(n^{O(\ell ^2)}\), the running time of the previously known algorithm obtained for graphs with bounded mim-width. We complement our result by showing that SFVS is \(\mathsf {W}\)[1]-hard parameterized by \(\ell \). 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 \(\mathsf {NP}\)-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.

Research supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research grant”, Project FANTA (eFficient Algorithms for NeTwork Analysis), number HFRI-FM17-431.

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Notes

  1. 1.

    The host tree is also known as a clique tree, usually when we are concerned with the maximal cliques of a chordal graph [19].

  2. 2.

    In this extended abstract all proofs are omitted due to space constraints. See a preliminary full version [36] for all the details.

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Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, Ioannina, Greece

    Charis Papadopoulos & Spyridon Tzimas

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  1. Charis Papadopoulos
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  2. Spyridon Tzimas
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Correspondence to Spyridon Tzimas .

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  1. Université Paris-Dauphine, Paris, France

    Cristina Bazgan

  2. Universität Trier, Trier, Germany

    Henning Fernau

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Papadopoulos, C., Tzimas, S. (2022). Computing a Minimum Subset Feedback Vertex Set on Chordal Graphs Parameterized by Leafage. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_34

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