Annals of Communications in Mathematics

Abstract:Let \( G \) be a connected graph. A non-empty set \( T \subseteq V(G) \) is a 2-movable total dominating set of \( G \) if \( T \) is a total dominating set and for every pair \( x,y \in T \), \( T \setminus \{x,y\} \) is a total dominating set in \( G \), or there exist \( u,v \in V(G) \setminus T \) such that \( u \) and \( v \) are adjacent to \( x \) and \( y \), respectively, and \( (T \setminus \{x,y\}) \cup \{u,v\} \) is a total dominating set in \( G \). The 2-movable total domination number of \( G \), denoted by \( \gamma_{mt}^{2}(G) \), is the minimum cardinality of a 2-movable total dominating set of \( G \). A 2-movable total dominating set with cardinality equal to \( \gamma_{mt}^{2}(G) \) is called a \( \gamma_{mt}^{2} \)-set of \( G \). This paper presents the 2-movable total domination in the join and corona of graphs.

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 \( W_{m,n} \).