Dom-Forcing Sets in Graphs and Its Relationships with Domination and Zero Forcing Number
P. Susanth
Department of Mathematics, Pookoya Thangal Memorial Government College, (Affiliated to University of Calicut), Perinthalmanna, Kerala- 679322, India and Department of Mathematics, St.Joseph’s College (Autonomous), Devagiri, Calicut - 673008, India.
Charles Dominic
Department of Mathematics, CHRIST (Deemed to be University), Bengaluru- 560029, India.
K. P. Premodkumar *
Department of Mathematics, Govt. College Malappuram, Kerala- 676509, India.
*Author to whom correspondence should be addressed.
Abstract
A dominating set Df ⊆ V (G) of a graph G is called a dom-forcing set if it also constitutes a zero forcing set. The minimum cardinality of such a set is defined as the dom-forcing number of G, denoted by Fd(G). This paper initiates a systematic study of the dom-forcing number and examines its relationship with classical graph parameters such as the domination number and the zero forcing number. We determine the exact values of dom-forcing number for several wellknown classes of graphs, including paths, cycles, coconut tree graphs, diamond snake graphs, triangular snake graphs, hypercubes, the Petersen graph, pineapple graphs, complete graphs, complete bipartite graphs, wheel graphs, and helm graphs. Further, we characterize classes of graphs for which the dom-forcing
number coincides with either the domination number or the zero forcing number. In addition, the dom-forcing number of the splitting graph of a given graph is investigated, and relevant bounds and structural properties are established. These results contribute to a deeper understanding of the interplay between domination and zero forcing processes in graphs.
Keywords: Zero forcing number, Domination number, Dom-forcing number