Dipayan Chakraborty ; Florent Foucaud ; Aline Parreau ; Annegret K. Wagler - On three domination-based identification problems in block graphs

fi:11003 - Fundamenta Informaticae, November 10, 2024, Volume 191, Issues 3-4: Iiro Honkala's 60 Birthday
On three domination-based identification problems in block graphsArticle

Authors: Dipayan Chakraborty ; Florent Foucaud ; Aline Parreau ; Annegret K. Wagler

    The problems of determining the minimum-sized \emph{identifying}, \emph{locating-dominating} and \emph{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 graphs classes. In this work, we present tight lower and upper bounds for all three types of codes for \emph{block graphs} (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or \emph{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.


    Volume: Volume 191, Issues 3-4: Iiro Honkala's 60 Birthday
    Published on: November 10, 2024
    Accepted on: May 22, 2024
    Submitted on: February 28, 2023
    Keywords: Mathematics - Combinatorics,Computer Science - Discrete Mathematics

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