k , l -Anonymity in Wheel-Related Social Graphs Measured on the Base of k -Metric Antidimension

Abstract

For the study and valuation of social graphs, which affect an extensive range of applications such as community decision-making support and recommender systems, it is highly recommended to sustain the resistance of a social graph $G$ to active attacks. In this regard, a novel privacy measure, called the $\left(k,l\right)$ -anonymity, is used since the last few years on the base of $k$ -metric antidimension of $G$ in which $l$ is the maximum number of attacker nodes defining the $k$ -metric antidimension of $G$ for the smallest positive integer $k$ . The $k$ -metric antidimension of $G$ is the smallest number of attacker nodes less than or equal to $l$ such that other $k$ nodes in $G$ cannot be uniquely identified by the attacker nodes. In this paper, we consider four families of wheel-related social graphs, namely, Jahangir graphs, helm graphs, flower graphs, and sunflower graphs. By determining their $k$ -metric antidimension, we prove that each social graph of these families is the maximum degree metric antidimensional, where the degree of a vertex is the number of vertices linked with that vertex.