Hyperdomination in Groups

Abstract

Hyperdomination in hypergraphs was defined by J. John Arul Singh and R. Kala in [3]. Let \(X = \{a_1, a_2, \ldots, a_n\}\) be a finite set and let \(\mathcal{E} = \{E_1, E_2, \ldots, E_m\}\) be a family of subsets of \(X\). \(H = (X, \mathcal{E})\)is said to be a hypergraph if (1) \(E_i \neq \phi\), \(1 \leq i \leq m\), and (2) \(\bigcup_{i=1}^{m} E_i = X\). The elements \(x_1, x_2, \ldots, x_n\) are called the vertices and the sets \(E_1, E_2, \ldots, E_m\) are called the edges. A set \(D \subset X\) is called a hyperdominating set if for each \(v \in X – D\) there exist some edge \(E\) containing \(v\) with \(|E| \geq 2\) such that \(E – v \subset D \neq D\). The hyperdomination number is the minimum cardinality of all hyperdominating sets. In this paper, a finite group is viewed as a hypergraph with vertex set as the elements of the group and edge set as the set of all subgroups of the group. We obtain several bounds for hyperdomination number of finite groups and characterise the extremal graphs in some cases.