The basis number of the strong product of trees and cycles with some graphs

The basis number of a graph \( G \) is defined to be the least integer \( d \) such that there is a basis \( \mathcal{B} \) of the cycle space of \( G \) such that each edge of \( G \) is contained in at most \( d \) members of \( \mathcal{B} \). MacLane [16] proved that a graph, \( G \), is planar if and only if the basis number of \( G \) is less than or equal to 2. Ali and Marougi [3] proved that the basis number of the strong product of two cycles and a path with a star is less than or equal to 4. In this work, (1) we prove the basis number of the strong product of two cycles is 3. (2) We give the exact basis number of a path with a tree containing no subgraph isomorphic to a 3-special star of order 7. (3) We investigate the basis number of a cycle with a tree containing no subgraph isomorphic to a 3-special star of order 7. The results in (1) and (2) improve the upper bound of the basis number of the strong product of two cycles and a star with a path which were obtained by Ali and Marougi.