Dom-forcing sets in graphs

Abstract

A dominating set $D_{f}\subseteq V(G)$ of vertices in a graph $G$ is called a \emph{dom-forcing set} if the sub-graph induced by $\langle D_{f} \rangle$ must form a zero forcing set. The minimum cardinality of such a set is known as the dom-forcing number of the graph $G$, denoted by $F_{d}(G)$. This article embarks on an exploration of the dom-forcing number of a graph $G$. Additionally, it delves into the precise determination of $F_{d}(G)$ for certain well-known graphs

Figures (10)

• Figure 1: In this graph $C_5$, $D_f=\{u_1, u_2, u_4\}$ is a dominating as well as zero forcing set and no subset of the vertex set with cardinality less than three has this property so $F_d(C_5)=3$
• Figure 2: $G(V,E)$
• Figure 3: In the cycle $C_{16}$, $S=\{v_1,v_2\}$ be a minimum zero forcing set and $G-N[S]$ is a path with vertex set $\{v_4, v_5, \cdots , v_{15}\}$. $T=\{v_5, v_8, v_{11}, v_{14}\}$ be a minimum dominating set of $G-N[S]$, therefore $\gamma(G-N[S])=4$. Hence $S \cup T$ is a dom-forcing set of $C_{16}$, which is minimum, $F_d(C_{16})=6$
• Figure 4: dom-forcing set for $P_3$ and $P_4$
• Figure 5: The Diamond Snake Graph $D_6$, dom-forcing set $D_f=\{ u_1, u_2, \cdots, u_6, v_2, v_4, v_6\}$ and $F_d(D_6)=9$

Theorems & Definitions (72)

• Proposition 1.2
• Example 1.3
• Proposition 1.4
• proof
• Theorem 2.1
• Example 2.2
• Theorem 2.3
• proof
• Theorem 2.5
• proof