, Volume 81, Issue 9, pp 3586–3629 | Cite as

Subset Feedback Vertex Set in Chordal and Split Graphs

  • Geevarghese Philip
  • Varun Rajan
  • Saket Saurabh
  • Prafullkumar TaleEmail author


In the Subset Feedback Vertex Set (Subset-FVS) problem the input is a graph G on n vertices, a subset T of vertices of G called the “terminal” vertices, and an integer k. The task is to determine whether there exists a subset of vertices of cardinality at most k which together intersect all cycles which pass through the terminals. Subset-FVS generalizes several well studied problems including Feedback Vertex Set and Multiway Cut. This problem is known to be NP-Complete, even in split graphs. Cygan et al. (SIAM J Discrete Math 27(1):290–309, 2013) proved that Subset-FVS is fixed parameter tractable (\(\mathsf {FPT}\)) in general graphs when parameterized by k. In split graphs a simple observation reduces the problem to an equivalent instance of the 3-Hitting Set problem with the same solution size. This directly implies, for Subset-FVSrestricted to split graphs, (i) an \(\mathsf {FPT}\) algorithm which solves the problem in \(\mathcal {O}^{\star } (2.076^k)\) time (The \(\mathcal {O}^{\star } ()\) notation hides polynomial factors.) (Wahlström in Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. Thesis, Department of Computer and Information Science, Linköpings universitet, 2007), and (ii) a kernel of size \(\mathcal {O}(k^3)\). We improve both these results for Subset-FVS on split graphs; we derive (i) a kernel of size \(\mathcal {O}(k^2)\) which is the best possible unless \(\textsf {NP}\subseteq {\mathsf {coNP}}/{\textsf {poly}}\), and (ii) an algorithm which solves the problem in time \(\mathcal {O}^*(2^k)\). Our algorithm, in fact, solves Subset-FVS on the more general class of chordal graphs, also in \(\mathcal {O}^*(2^k)\) time. To the best of our knowledge, the fastest known exact algorithm for Subset-FVS on chordal graphs is based on the 3-Hitting Set algorithm of Fomin et al. (JACM 66(2):8, 2019) which runs in \(\mathcal {O}^*(1.5182^n)\) time. Applying the results of Fomin et al. (2019) to our \(\mathsf {FPT}\) algorithm yields two exact exponential-time algorithms for Subset-FVS on chordal graphs: a randomized algorithm which runs in \(\mathcal {O}^*(1.5^{n})\) time, and a deterministic algorithm which runs in \(\mathcal {O}^*((1.5+\varepsilon )^{n})\) time for any fixed \(\varepsilon >0\).


Subset feedback vertex set Chordal and split graphs Parameterized complexity 



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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Geevarghese Philip
    • 1
  • Varun Rajan
    • 2
  • Saket Saurabh
    • 3
    • 4
  • Prafullkumar Tale
    • 5
    Email author
  1. 1.Chennai Mathematical Institute and UMI ReLaXChennaiIndia
  2. 2.Chennai Mathematical InstituteChennaiIndia
  3. 3.The Institute Of Mathematical Sciences, HBNI and UMI ReLaXChennaiIndia
  4. 4.Department of InformaticsUniversity of BergenBergenNorway
  5. 5.The Institute Of Mathematical Sciences, HBNIChennaiIndia

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