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.

翻译：图的完美匹配的强迫集定义为该完美匹配的一个边子集，使得存在唯一的完美匹配包含该子集。图的完全强迫集是其边的一个子集，使得对于图的每个完美匹配，该子集与该完美匹配的某个强迫集相交。图的最小完全强迫集的大小称为该图的完全强迫数。本文中，我们根据其他图论参数（如图的退化度或谱半径）推导出图的完全强迫数的新上界。我们证明，对于边数超过顶点数某个常数倍的图，我们的结果优于已知的完全强迫数最佳上界。对于边传递图集，我们给出了完全强迫数关于最大强迫数的下界。该结果特别应用于超立方图以及偶环的笛卡尔幂。