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 $G$ which is uniquely extendable to a perfect matching in $G$. Diwan (2019) proved by linear algebra that for $d$-hypercube $Q_d$ ($d\ge 2)$, $f(Q_n)=2^{d-2}$, settling a conjecture proposed by Pachter and Kim in 1998. Recently Mohammadian generalized this method to obtain a general result: for a bipartite graph $G$ on $n$ vertices, if there exists an involutory matrix $A$ on a field $F$ as a weighted adjacency matrix then $f(G\mathrm{\square }{K}_2)=\frac{n}{2}$. In this paper, under the same condition we obtain $f(G\mathrm{\square }{C}_{2k})=n(k\ge 2)$. Also this method can be applied to some non-balanced bipartite graphs $G$ whenever $G$ admit a weighted bi-adjacency matrix with orthogonal rows.