The notion of forcing number for a perfect matching or innate degree of freedom for a Kekulé structure was introduced by Randić and Klein in 1980s, which plays an important role in the resonance theory in organic chemistry. Over the past four decades, some theoretical chemists and mathematicians have been attracted to investigate the effects in some chemical graphs and the graph-theoretical problems in general graphs. Recently some derived concepts and related problems have also been raised successively, such as anti-forcing number of a perfect matching, global forcing number, complete forcing number, etc. This survey will present recent progress on the minimum and maximum (anti-) forcing numbers, (anti-) forcing spectra and (anti-) forcing polynomials of perfect matchings of graphs, and global forcing and complete forcing numbers with their relations, as well as some open problems and conjectures.

Abstract For connected graphs $G$ and $H$, the $H$-structure connectivity $\kappa (G; H)$ (resp. $H$-substructure connectivity $\kappa ^{s}(G; H)$) of $G$ is the minimum cardinality of a set of subgraphs $\mathcal{F}$ of $G$ such that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so that $G-\mathcal{F}$ is disconnected or singleton. In this paper, we consider $P_t$-structure connectivity and $P_t$-substructure connectivity of augmented $k$-ary $n$-cubes $AQ_{n,k}$ for $n\geq 2$, $k\geq 3$ and $1\leq t\leq 4n-2$. We obtain that $\kappa (AQ_{n,k}; P_t)=\kappa ^s(AQ_{n,k}; P_t)=\frac{4n-2}{t}+1$ for $t\mid 4n-2$, $t\nmid 2n-1$, $t>6$, $n\geq 3$ and $k\geq 4$; $\kappa (AQ_{n,k}; P_t)=\kappa ^s(AQ_{n,k}; P_t)=\lceil \frac{4n-2}{t}\rceil $, in other cases.