Induced Cycle Path Number of Graphs

Let G=(V, E) be a simple connected graph. An induced cycle path partition of G is a partition of V (G) into subsets such that the subsets induce cycles and paths. The minimum order taken over all induced cycle path partitions is called the induced cycle path number of G and is denoted by ρcp(G). In this paper, we dertermine this parameter for some standard graphs and investigate the same for unicyclic graphs and Halin graphs. We also obtain an upper bound for arbitrary graphs and characterize the extremal graphs. The graphs which have unique induced cycle path partition are also discussed