Abstract:

A connected dominating set \( C \) in \( G \) is a 2-movable connected dominating set of \( G \) if for every pair \( x,y \in C \), \( C \setminus \{x,y\} \) is a connected dominating set in \( G \), or there exists \( u,v \in V(G) \setminus C \) such that \( u \) and \( v \) are adjacent to \( x \) and \( y \), respectively, and \( (C \setminus \{x,y\}) \cup \{u,v\} \) is a connected dominating set of \( G \). The minimum cardinality of a 2-movable connected dominating set of \( G \), denoted by \( \gamma_{mc}^{2}(G) \) is the 2-movable connected domination number of \( G \). A 2-movable connected dominating set with cardinality \( \gamma_{mc}^{2}(G) \) is called a minimum 2-movable connected dominating set or a \( \gamma_{mc}^{2} \)-set of \( G \). In this paper, the researcher obtained the 2-movable connected domination number of the Complete Graph \( K_n \), Join Graph \( G + H \), and Generalized Wheel Graph \( W_{m,n} \).