Connected dom-forcing sets in graphs

TL;DR

The work introduces the connected dom-forcing number $F_{cd}(G)$, marrying connected domination with connected zero forcing to study minimal starter sets that simultaneously dominate and force an entire graph. It establishes core bounds in terms of $Z_c(G)$ and $\gamma_c(G)$, and explores exact values for foundational graphs and a wide class of graph constructions, including joins, corona and rooted products, and Cartesian products, as well as splitting graphs. The results yield precise formulas and tight bounds for many families (paths, cycles, grids, ladders, and various products), revealing both additive and multiplicative behavior under graph composition. These findings advance understanding of robust, connected influence spread in networks and pose new questions about subgraph preservation and ladder graph behavior.

Abstract

In a graph G, a dominating set Df subset of V (G) is called a dom-forcing set if the sub-graph induced by Df 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 Fd(G). A connected dom-forcing forcing set of a graph G, is a dom-forcing set of G that induces a sub graph of G which is connected. The connected dom-forcing number of G, Fcd(G), is the minimum size of a connected dom-forcing set. This study delves into the concept of the connected dom-forcing number Fcd(G), examining its properties and characteristics. Furthermore, it seeks to accurately determine Fcd(G) for several well-known graphs and their graph products.

Figures (7)

• Figure 1: In this graph $C_5$, $CD_f=\{u_1, u_2, u_3\}$ is a connected dominating as well as zero forcing set and no subset of the vertex set with cardinality less than three has this property so $F_{cd}(C_5)=3$
• Figure 2: The wheel graph $W_{16}$
• Figure 3: The Cycle $C_{16}$
• Figure 4: The Graph G
• Figure 5: The Graph H

Theorems & Definitions (59)

• Definition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7
• proof
• Proposition 3.8
• Proposition 3.9