Annals of Communications in Mathematics

Abstract:An injective function \( f : V(G) \to \{T_0, T_1, T_2, \ldots, T_n\} \), where \( n = |V(G)| - 1 \), is said to be a Tribonacci cordial labeling if the induced function \( f^{*} : E(G) \to \{0,1\} \) defined by \( f^{*}(uv) = (f(u) + f(v)) \pmod 2 \) satisfies the condition \( |e_f(0) - e_f(1)| \le 1 \), where \( e_f(0) \) is the number of edges with label \( 0 \) and \( e_f(1) \) is the number of edges with label \( 1 \). A graph that admits such labeling is called a Tribonacci cordial graph. In this paper, we determine the Tribonacci cordial labeling of Triangular Snake Graph \( TS_n \), Double Triangular Snake Graph \( DT(S_n) \), Quadrilateral Snake Graph \( QS_n \), Double Quadrilateral Snake Graph \( D(QS_n) \), and Cycle Quadrilateral Snake Graph \( C(QS_n) \).

Abstract:A dominating set \( S \) in \( G \) is called a pendant dominating set if \( \langle S \rangle \) contains at least one pendant vertex. The minimum cardinality of a pendant dominating set is called the pendant domination number, denoted by \( \gamma_{pe}(G) \). The pendant domination polynomial of \( G \) is denoted by \( D_{pe}(G,x) \) and is defined as\[D_{pe}(G,x)=\sum_{i=\gamma_{pe}(G)}^{n} d_{pe}(G,i)\,x^{i},\]where \( d_{pe}(G,i)x^{i} \) is the number of pendant dominating sets of size \( i \). In this paper, we obtained the pendant domination number and pendant domination polynomial of the corona of some graphs, namely, \( P_m \circ K_n \), \( C_m \circ K_n \), and \( K_m \circ K_n \).

Abstract:An injective function \( f : V(G) \to \{L_1, L_2, \ldots, L_n\} \), where \( L_j \) is the \( j^{\text{th}} \) Lucas number \( (j=1,2,\ldots,n) \), is said to be a Lucas cordial labeling if the induced function \( f^{*} : E(G) \to \{0,1\} \) defined by \( f^{*}(uv) = (f(u)+f(v)) \pmod 2 \) satisfies \( |e_f(0)-e_f(1)| \le 1 \). A graph admitting such labeling is called a Lucas cordial graph.

Abstract:A total product cordial labeling of a graph \( G \) is a function \( f : V \rightarrow \{0,1\} \). For each \( xy \), assign the label \( f(x)f(y) \); \( f \) is called total product cordial labeling of \( G \) if it satisfies the condition that \( |v_f(0)+e_f(0)-v_f(1)-e_f(1)| \leq 1 \) where \( v_f(i) \) and \( e_f(i) \) denote the set of vertices and edges which are labeled with \( i = 0,1 \), respectively. A graph with a total product cordial labeling defined on it is called a total product cordial graph. In this paper, we determined the total product cordial labeling of the snake graphs \( T_n \), \( A(T_n) \), \( D(T_n) \), \( DA(T_n) \), \( Q_n \), \( A(Q_n) \), \( D(Q_n) \), and \( DA(Q_n) \).

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 \( W_{m,n} \).