Advances and Applications in Discrete Mathematics

A zero forcing set in a graph is called a dom-forcing set if is also a dominating set of . The minimum cardinality of such a set is known as the dom-forcing number of the graph , denoted by . If the subgraph induced by a dom-forcing set is connected, then it is called a connected dom-forcing set of the graph . The minimum cardinality of such a set is called the connected domforcing number of , denoted by . The propagation time of a connected dom-forcing set of a graph is the minimum number of steps required to force all vertices to become black, starting from the vertices in the connected dom-forcing set and performing independent forces simultaneously. The connected dom-forcing propagation time of a graph is the minimum of propagation time taken over all minimum connected dom-forcing sets of the graph. We discuss the minimum connected dom-forcing propagation time of the graph . Additionally, it delves into the precise determination of minimum connected dom-forcing propagation time for certain well-known graphs. Also, we characterize the class of graphs that admit connected dom-forcing propagation times and .