1-Factor Forcing in Certain Nanotori

Abstract

Let S be a set of vertices of a graph G which are coloured in black, whilst all other vertices remain non-coloured. A black vertex with exactly one non-coloured neighbour forces the neighbour to be coloured black at each discrete time step. By iteratively applying the method, if all vertices get coloured black at some stage, then the set S is called a zero forcing set of G. If S induces a perfect matching or a 1-factor, then S is termed a 1-factor forcing set of G. The 1-factor forcing number of G, denoted by ζP2(G), is the minimum cardinality of a 1-factor forcing set of G. The 1-factor forcing problem in a graph G is to determine 1-factor forcing set of G of minimum cardinality. In this paper, we solve the 1-factor forcing problem for certain nanotori such as H-Naphtelanic [m, n] and C4C6C8[ m, n] Nanotori.