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Minimal Roman Dominating Functions: Extensions and Enumeration

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Graph-Theoretic Concepts in Computer Science (WG 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13453))

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Abstract

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We develop non-trivial enumeration algorithms for minimal Roman dominating functions with polynomial delay and polynomial space. Recall that the existence of a similar enumeration result for minimal dominating sets is open for decades. Our result is based on a polynomial-time algorithm for Extension Roman Domination: Given a graph \(G=(V,E)\) and a function \(f:V\rightarrow \{0,1,2\}\), is there a minimal Roman dominating function \(\tilde{f}\) with \(f\le \tilde{f}\)? Here, \(\le \) lifts \(0< 1< 2\) pointwise; minimality is understood in this order. Our enumeration algorithm is also analyzed from an input-sensitive viewpoint, leading to a run-time estimate of \(\mathcal {O}(1.9332^n)\) for graphs of order n; this is complemented by a lower bound example of \(\varOmega (1.7441^n)\).

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Notes

  1. 1.

    Historically, a logical extension problem [10] should be mentioned, as it has led to [40, Théorème 2.16], dealing with an extension variant of 3-Hitting Set; also see [40, Proposition 3.39] concerning implications for Extension Dominating Set.

  2. 2.

    According to [29], this notion of minimality for rdf was coined by Cockayne but then dismissed, as it does not give a proper notion of upper Roman domination. However, in our context, this definition seems to be the most natural one, as it also perfectly fits the extension framework proposed in [13]; see the discussions in Sect. 7.

  3. 3.

    This condition assumes that our graphs have non-empty vertex sets.

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Author information

Authors and Affiliations

  1. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Faisal N. Abu-Khzam

  2. Fachbereich 4 – Abteilung Informatikwissenschaften, Universität Trier, 54286, Trier, Germany

    Henning Fernau & Kevin Mann

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  1. Faisal N. Abu-Khzam
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  2. Henning Fernau
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Correspondence to Henning Fernau .

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  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Universität Tübingen, Tübingen, Germany

    Michael Kaufmann

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Abu-Khzam, F.N., Fernau, H., Mann, K. (2022). Minimal Roman Dominating Functions: Extensions and Enumeration. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_1

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