Minimum forcing numbers of perfect matchings of circular and prismatic graphs

Let G be a graph with a perfect matching. Denote by 𝑓 ⁡ ( 𝐺 ) the minimum size of a matching in G that is uniquely extendable to a perfect matching in G . Diwan (2019) used linear algebra to prove that for the d -hypercube 𝑄 𝑑 ( 𝑑 ≥ 2 ) , 𝑓 ⁡ ( 𝑄 𝑑 ) = 2 𝑑 − 2 , thus settling a conjecture of Pachter and Kim (1998). Recently, Mohammadian generalized this method to prove a general result: for a bipartite graph G on n vertices, if G admits an involutory weighted adjacency matrix A over a field F , then 𝑓 ⁡ ( 𝐺 □ 𝐾 2 ) = 𝑛 2 , where □ denotes the Cartesian product of two graphs. In this paper we obtain 𝑓 ⁡ ( 𝐺 □ 𝐶 2 ⁢ 𝑘 ) = 𝑛 when a bipartite graph G on n vertices admits an involutory weighted adjacency matrix A over a field F of characteristic not 2, for all integers 𝑘 ≥ 2 . Moreover, we demonstrate that this method can also be applied to some unbalanced bipartite graphs G when G admits a weighted bi-adjacency matrix with orthogonal rows.