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International Workshop on Combinatorial Algorithms

IWOCA 2022: Combinatorial Algorithms pp 3–20Cite as

Distance from Triviality 2.0: Hybrid Parameterizations

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

Abstract

Vertex deletion problems have been at the heart of numerous major advances in Algorithms and Combinatorial Optimization, and especially so in the area of Parameterized Complexity. For a family of graphs \(\mathcal H\), the input to Vertex Deletion to \(\mathcal {H}\) is a graph G and an integer k, and the objective is to decide whether there is a vertex-subset, called a modulator, whose removal from G results in a graph contained in the family \(\mathcal H\), and such that \(|S|\le k\). Traditionally, the majority of the study of Vertex Deletion to \(\mathcal {H}\) problems in Parameterized Complexity has been limited to parameterization by modulator size and structural graph width measures of the input graph such as treewidth. Recent years have seen systematic efforts at: i) quantifying the complexity of modulators in ways other than their size, and ii) studying the complexity landscape of various graph problems under parameterizations that are simultaneously better than both the modulator size and certain width measures of the graph. In this talk we will look at some exciting developments in this direction in relation to two such parameters that are “hybridizations” of the modulator size, and the well-explored graph parameters – treewidth and treedepth.

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Notes

  1. 1.

    For this, we always assume that \(\mathcal {H}\) contains the empty graph and so V(G) is a trivial modulator to \(\mathcal {H}\).

  2. 2.

    Here we make a mild assumption that, checking whether a graph is in \(\mathcal H\) can be done in polynomial time.

  3. 3.

    A family of graphs \(\mathcal H\) is hereditary if for each graph \(G \in \mathcal{H}\), every induced subgraph of G belongs to \(\mathcal H\).

  4. 4.

    When we say CMSO, we refer to the fragment that is sometimes referred to as \({\mathsf{CMSO}}_2\) in the literature. We will only be using a meta-result regarding CMSO definable problems, and thus, we refer to [5, 11, 12] for a an introduction to this topics.

  5. 5.

    We can check if \(G \in \mathcal{H}\) by calling \(\mathscr {X}_\mathsf{mod}\) for the instance (G, 0). We recall that \(\mathcal H\) is closed under disjoint union.

  6. 6.

    Even if \(\mathscr {X}_\mathsf{mod}\) is a decision algorithm, using the self-reducibility like property, we can compute the decomposition itself, see Lemma 3.5 and 3.6 in [1] or [2] for details.

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Authors and Affiliations

  1. Indian Institute of Technology Madras, Chennai, India

    Akanksha Agrawal

  2. University of Warwick, Coventry, England

    M. S. Ramanujan

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  1. Akanksha Agrawal
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  2. M. S. Ramanujan
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Correspondence to Akanksha Agrawal .

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  1. Université Paris-Dauphine, Paris, France

    Cristina Bazgan

  2. Universität Trier, Trier, Germany

    Henning Fernau

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Agrawal, A., Ramanujan, M.S. (2022). Distance from Triviality 2.0: Hybrid Parameterizations. 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_1

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