CONNECTED FORCING NUMBER OF CERTAIN GRAPHS

Given a simple graph $ G=(V,E) $ with a set $ S \subseteq V $, to be an initial set of coloured vertices called black vertices and all remaining vertices being uncoloured, called white vertices. At each integer valued time step, a coloured vertex in the set $ S $ with a single uncoloured neighbour will force that neighbour to get coloured and such a vertex is called a forcing vertex and the set $ S $ is called a forcing set, if by relatively applying the forcing process, all of $ V $ becomes coloured. The forcing number of a graph $ G $ is the cardinality of the smallest forcing set of $ G $ and it is denoted by $ F(G) $. One of the variants of forcing, namely connected forcing, is a restriction of forcing in which initial set of coloured vertices induces a connected subgraph. The connected forcing number, $ F_{c}(G) $ of a graph $ G $, is the minimum cardinality among all connected forcing sets of $ G $. In this paper, we determine $ F_{c}(G) $ of degree splitting graphs and line graphs of certain graphs. Further we discuss on its bounds and the realizability theorem.