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Parameterized Complexity of Generalizations of Edge Dominating Set

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SOFSEM 2025: Theory and Practice of Computer Science (SOFSEM 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15538))

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Abstract

The objective of this article is to propose two natural generalizations of covering edges by edges (Edge Dominating Set) and study these problems from the multivariate lens. The first is simply considering Edge Dominating Set on hypergraphs, called Hyperedge Dominating Set. Given a hypergraph \(\mathcal{H}=( \mathcal U,\mathcal{F})\), a set \(F \subseteq \mathcal{F}\) is called a hyperedge dominating set if all hyperedges intersect with at least one hyperedge \(e \in F\). The objective of the Hyperedge Dominating Set problem is to determine whether a hyperedge dominating set of size at most k exists. We find it quite surprising that such generalization is missing from the literature. The second extension we consider is the t-Path Edge Dominating Set problem. In this problem, the input consists of a graph G and an integer k, and the goal is to find a set \(\mathcal{P}\) of at most k paths, each of length at most t, such that for every edge in G, at least one of its endpoints belongs to the vertex set V(P) for some \(P \in \mathcal{P}\). We show the following results and add to the literature on Edge Dominating Set.

  • Hyperedge Dominating Set is FPT parameterized by \(k+d\), where d is the maximum size of a hyperedge in the input hypergraph.

  • A kernel of size \(\mathcal {O}(k^d)\) can be obtained for the Hyperedge Dominating Set problem, where d is the maximum size of a hyperedge in the input hypergraph.

  • The problem of finding a Hyperedge Dominating Set is computationally difficult; specifically it is W[2]-hard when parameterized by k. This hardness result holds even when each vertex is contained in at most 2 hyperedges and the intersection between any two hyperedges is at most 1.

  • t-Path Edge Dominating Set is FPT when parameterized by \(k+t\). Additionally, it has a kernel of size \(\mathcal {O}(k^3t^3)\).

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Notes

  1. 1.

    The \({\mathcal O^{\star }}\) notation hides polynomial factors.

  2. 2.

    Proofs of results marked with \(\star \) are omitted due to paucity of space.

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, IIT Hyderabad, Kandi, India

    Shubhada Aute

  2. School of Computer Science, University of Leeds, Leeds, UK

    Fahad Panolan

  3. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Souvik Saha, Saket Saurabh & Anannya Upasana

  4. University of Bergen, Bergen, Norway

    Saket Saurabh

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  1. Shubhada Aute
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  4. Saket Saurabh
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  5. Anannya Upasana
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Correspondence to Shubhada Aute .

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  1. Comenius University in Bratislava, Bratislava, Slovakia

    Rastislav Královič

  2. Czech Academy of Sciences, Prague, Czech Republic

    Věra Kůrková

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Aute, S., Panolan, F., Saha, S., Saurabh, S., Upasana, A. (2025). Parameterized Complexity of Generalizations of Edge Dominating Set. In: Královič, R., Kůrková, V. (eds) SOFSEM 2025: Theory and Practice of Computer Science. SOFSEM 2025. Lecture Notes in Computer Science, vol 15538. Springer, Cham. https://doi.org/10.1007/978-3-031-82670-2_6

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