We begin research on a variant of standard domination known as restrained domination. Consider the graph � = (�, �). When �[�] = � (�), a subset � of � (�) is said to dominate � if and only if for any � ∈ (� (�)\�) there is a vertex in � such that �� ∈ �(�). A restrained dominating set is a set � ⊆ � in which each vertex in � − � is adjacent to both A vertex in � and another vertex in � − �. The smallest cardinality of a restrained dominating set of � is denoted by the restrained domination number of �, denoted by ��(�). For some positive integers � and � such that � �� [1, 2, . . . , � − 3, � − 2] and � ≥ 4 , there Exists a connected nontrivial graph � with |� (�)| = � and ��−1 (�) = � Also, with |�1(�)| = 1 ��� ����(�) = 3 . Then|�(�)| + 2 ≤ ���(�) ≤ ���(�) + |�| + 1, where � ∈ �1(�) ��� � = {�|�(�) ⊆ �(�)} − {�|�(�[�[�] − {�}]) ⊆ �[�]}
