Independent Italian bondage of graphs

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Kosari Saeed; Amjadi Jafar; Khan Aysha; Volkmann Lutz

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Journal Paper

Paper Information

Journal: COMMUNICATIONS IN COMBINATORICS AND OPTIMIZATION Year:2023 | Volume:8 | Issue:4 Page(s): 649-664

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Information Journal Paper

Title

Independent Italian bondage of graphs

Author(s)

Kosari Saeed | Amjadi Jafar | Khan Aysha | Volkmann Lutz | Issue Writer Certificate

Keywords

Independent Italian dominating function

independent Italian domination number

independent Italian bondage number

Abstract

An independent Italian dominating function (IID-function) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the conditions that (i) $\sum_{u\in N(v)}f(u)\geq2$ when $f(v)=0$, and (ii) the set of all vertices assigned non-zero values under $f$ is independent. The weight of an IID-function is the sum of its function values over all vertices, and the independent Italian domination number $i_{I}(G)$ of $G$ is the minimum weight of an IID-function on $G$. In this paper, we initiate the study of the independent Italian bondage number $b_{iI}(G)$ of a graph $G$ having at least one component of order at least three, defined as the smallest size of a set of edges of $G$ whose removal from $G$ increases $i_{I}(G)$. We show that the decision problem associated with the independent Italian bondage problem is NP-hard for arbitrary graphs. Moreover, various upper bounds on $b_{iI}(G)$ are established as well as exact values on it for some special graphs. In particular, for trees $T$ of order at least three, it is shown that $b_{iI}(T)\leq2$.

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