Forcing a unique minimum spanning tree and a unique shortest path

A forcing set $S$ in a combinatorial problem is a set of elements such that there is a unique solution that contains all the elements in $S$. An anti-forcing set is the symmetric concept: a set $S$ of elements is called an anti-forcing set if there is a unique solution disjoint from $S$. There are extensive studies on the computational complexity of finding a minimum forcing set in various combinatorial problems, and the known results indicate that many problems would be harder than their classical counterparts: finding a minimum forcing set for perfect matchings is NP-hard [Adams et al., Discret. Math. 2004] and finding a minimum forcing set for satisfying assignments for 3CNF formulas is $\mathrm{\Sigma}_2^P$-hard [Hatami-Maserrat, DAM 2005]. In this paper, we investigate the complexity of the problems of finding minimum forcing and anti-forcing sets for the shortest $s$-$t$ path problem and the minimum weight spanning tree problem. We show that, unlike the aforementioned results, these problems are tractable, with the exception of finding a minimum anti-forcing set for shortest $s$-$t$ paths, which is NP-hard.