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Exploring the Gap Between Treedepth and Vertex Cover Through Vertex Integrity

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Algorithms and Complexity (CIAC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12701))

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Abstract

For intractable problems on graphs of bounded treewidth, two graph parameters treedepth and vertex cover number have been used to obtain fine-grained complexity results. Although the studies in this direction are successful, we still need a systematic way for further investigations because the graphs of bounded vertex cover number form a rather small subclass of the graphs of bounded treedepth. To fill this gap, we use vertex integrity, which is placed between the two parameters mentioned above. For several graph problems, we generalize fixed-parameter tractability results parameterized by vertex cover number to the ones parameterized by vertex integrity. We also show some finer complexity contrasts by showing hardness with respect to vertex integrity or treedepth.

Partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP18K11168, JP18K11169, JP19K21537, JP20K19742, JP20H05793.

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Notes

  1. 1.

    For example, by fixing the ordering of vertices in S as \(v_{1}, \dots , v_{|S|}\), we can set t to be the adjacency matrix of \(G[S \cup V(C)]\) such that the ith row and column correspond to \(v_{i}\) for \(1 \le i \le |S|\) and under this condition the string \(t[1,1], \dots , t[1, s], t[2,1], \dots , t[s,s]\) is lexicographically minimal, where \(s = |S \cup V(C)|\).

  2. 2.

    In [40], W[1]-hardness was stated for \(\mathsf {tw}\) but the proof shows it for \(\mathsf {td}\) as well.

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Authors and Affiliations

  1. Nagoya University, Nagoya, Japan

    Tatsuya Gima, Tesshu Hanaka & Yota Otachi

  2. Seikei University, Musashino-shi, Tokyo, Japan

    Masashi Kiyomi

  3. Kyoto University, Kyoto, Japan

    Yasuaki Kobayashi

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  1. Tatsuya Gima
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Correspondence to Yota Otachi .

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  1. Sapienza University of Rome, Rome, Italy

    Tiziana Calamoneri

  2. Sapienza University of Rome, Rome, Italy

    Federico Corò

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Gima, T., Hanaka, T., Kiyomi, M., Kobayashi, Y., Otachi, Y. (2021). Exploring the Gap Between Treedepth and Vertex Cover Through Vertex Integrity. In: Calamoneri, T., Corò, F. (eds) Algorithms and Complexity. CIAC 2021. Lecture Notes in Computer Science(), vol 12701. Springer, Cham. https://doi.org/10.1007/978-3-030-75242-2_19

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