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On Three Domination-Based Identification Problems in Block Graphs

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Algorithms and Discrete Applied Mathematics (CALDAM 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13947))

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Abstract

The problems of determining the minimum-sized identifying, locating-dominating and open locating-dominating codes of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set C of a graph G such that the vertices of a chosen subset of V(G) (i.e. either \(V(G)\setminus C\) or V(G) itself) are uniquely determined by their neighborhoods in C. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graph classes. In this work, we present tight lower and upper bounds for all three types of codes for block graphs (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or blocks) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature - with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.

This work was sponsored by a public grant overseen by the French National Research Agency as part of the “Investissements d’Avenir” through the IMobS3 Laboratory of Excellence (ANR-10-LABX-0016) and the IDEX-ISITE initiative CAP 20-25 (ANR-16-IDEX-0001). We also acknowledge support of the ANR project GRALMECO (ANR-21-CE48-0004).

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Authors and Affiliations

  1. Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000, Clermont-Ferrand, France

    Dipayan Chakraborty, Florent Foucaud & Annegret K. Wagler

  2. Univ Lyon, CNRS, INSA Lyon, UCBL, Centrale Lyon, Univ Lyon 2, LIRIS, UMR5205, F-69622, Villeurbanne, France

    Aline Parreau

Authors
  1. Dipayan Chakraborty
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  2. Florent Foucaud
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  3. Aline Parreau
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  4. Annegret K. Wagler
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Corresponding author

Correspondence to Florent Foucaud .

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Editors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology Delhi, New Delhi, India

    Amitabha Bagchi

  2. Dhirubhai Ambani Institute of Information and Communication Technology, Gandhinagar, India

    Rahul Muthu

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Chakraborty, D., Foucaud, F., Parreau, A., Wagler, A.K. (2023). On Three Domination-Based Identification Problems in Block Graphs. In: Bagchi, A., Muthu, R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2023. Lecture Notes in Computer Science, vol 13947. Springer, Cham. https://doi.org/10.1007/978-3-031-25211-2_21

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  • DOI: https://doi.org/10.1007/978-3-031-25211-2_21

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