Bounds on the Complete Forcing Number of Graphs
Javad B. Ebrahimi, Aref Nemayande, Elahe Tohidi
TL;DR
The paper studies the complete forcing number $cf(G)$, the size of the smallest subset of edges that intersects every forcing set across all perfect matchings. It introduces a constructive vertex-ordering algorithm to produce complete forcing sets and derives sharp upper bounds in terms of the spectral radius $\rho(G)$ and degeneracy $d$ (specifically $cf(G) \le \left(1 - \frac{1}{\rho(G)}\right)|E(G)|$ and $cf(G) \le \left(1 - \frac{1}{2\sqrt{d\Delta} - d}\right)|E(G)|$). The paper also establishes a universal lower bound for edge-transitive graphs: $cf(G) \ge \frac{2|E(G)|}{|V(G)|} F(G)$, yielding concrete estimates for hypercube graphs $Q_n$ and Cartesian powers $C_n^k$ with even $n$. By connecting $cf(G)$ to spectral and degeneracy parameters, the results provide computable bounds applicable to planar, outerplanar, and product graphs, and they extend the understanding of forcing structures in chemical graph models. The work advances both theory and practical bound computation for the complete forcing number across broad graph families.
Abstract
A forcing set for a perfect matching of a graph is defined as a subset of the edges of that perfect matching such that there exists a unique perfect matching containing it. A complete forcing set for a graph is a subset of its edges, such that it intersects the edges of every perfect matching in a forcing set of that perfect matching. The size of a smallest complete forcing set of a graph is called the complete forcing number of the graph. In this paper, we derive new upper bounds for the complete forcing number of graphs in terms of other graph theoretical parameters such as the degeneracy or the spectral radius of the graph. We show that for graphs with the number of edges more than some constant times the number of vertices, our result outperforms the best known upper bound for the complete forcing number. For the set of edge-transitive graphs, we present a lower bound for the complete forcing number in terms of maximum forcing number. This result in particular is applied to the hypercube graphs and Cartesian powers of even cycles.