On the basis number of the composition of different ladders with some graphs

The graphs considered in this paper are finite, undirected, simple, and connected. Most of the notations that follow can be found in [6] or [8]. Let G= (V ,E) be a graph, where V and E are the vertex and the edge sets of G, respectively. If e1,e2, . . . ,eq is an ordering of the edges in G, then any subset S of edges corresponds to a (0,1)-vector (a1,a2, . . . ,aq) in the usual way, with ai = 1 (ai = 0) if and only if ei ∈ S (ei / ∈ S). These vectors form a qdimensional vector space (Z2) over the field Z2. The vectors in (Z2) which correspond to the cycles in G generate a subspace called the cycle space of G denoted by (G). We will say that the cycles themselves, rather than the vectors corresponding to them, generate (G). It is known that for a connected graph G, dim (G) = ∣E(G)∣−∣V(G)∣+ 1. (1.1)