Minimum forcing numbers of perfect matchings of circular and prismatic graphs

TL;DR

This work addresses the minimum forcing number, $f(G)$, of perfect matchings, focusing on circular and prismatic graph products. It extends the algebraic approach based on involutory weighted adjacency matrices to derive exact values of $f(G\Box C_{2k})$ and $f(G\Box K_2)$ under suitable conditions, and it extends to non-balanced bipartite graphs via weighted bi-adjacency matrices with orthogonal or unitary rows. The main contributions include proving $f(G\Box C_{2k})=n$ for bipartite $G$ with $\mathcal{I}_F(G)\neq\emptyset$ and $k\ge2$, and proving $f(G\Box K_2)=m$ whenever $|X|=m\le n$ and there exist $B,C$ with $BC^{\top}=I_m$, along with several illustrative examples. Overall, the paper provides a unified linear-algebraic framework to compute forcing numbers for circular and prism graph constructions, enriching both combinatorial and chemical graph theory contexts.

Abstract

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 $Q_d$ ($d\geq 2)$, $f(Q_n)=2^{d-2}$, 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 $f(G\Box K_2)=\frac{n}{2}$. In this paper, under the same condition we obtain $f(G\Box C_{2k})=n ~(k\ge2)$. Also this method can be applied to some non-balanced bipartite graphs $G$ whenever $G$ admit a weighted bi-adjacency matrix with orthogonal rows.

Figures (4)

• Figure 1: Signed graph $S_{14}$ with a weighted bi-adjacency matrix (see also Mckee).
• Figure 2: Illustration for graphs $G_1$ (left) and $G_2$ (right).
• Figure 3: $(0,1,-1)$-matrices $E(4t+2,5)$ (left) and $F(4t+4,5)$ (right), $t\geq 2$ (taken from stani).
• Figure 4: A bipartite graph $G'$ with a weighted bi-adjacency matrix (see Brennan).

Theorems & Definitions (20)

• Theorem 1.1: Diwan Diwan
• Lemma 1.2: Diwan Diwan
• Lemma 1.3: Diwan Diwan, Mohammadian Mohammadian
• Theorem 1.4: Mohammadian Mohammadian
• Theorem 2.1
• proof
• Corollary 2.2
• Remark 2.3
• Theorem 3.1
• proof