Forcing a unique minimum spanning tree and a unique shortest path

Tatsuya Gima; Andreas Grigorjew; Yasuaki Kobayashi; Michael Lampis; Yiren Lu; Valia Mitsou; Edouard Nemery; Yuto Okada; Yota Otachi; Takumi Sato

arXiv · 2509.24309

Forcing a unique minimum spanning tree and a unique shortest path

Tatsuya Gima·Andreas Grigorjew·Yasuaki Kobayashi·Michael Lampis·Yiren Lu·Valia Mitsou·Edouard Nemery·Yuto Okada·Yota Otachi·Takumi Sato

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Abstract

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 are harder than their classical counterparts: the decision version of finding a minimum forcing set for perfect matchings is NP-complete [Adams et al., Discrete Mathematics, 2004], and that of finding a minimum forcing set for satisfying assignments for 3CNF formulas is $\Sigma_2^P$-complete [Hatami-Maserrat, Discrete Applied Mathematics, 2005]. In this paper, we investigate the complexity 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 the decision version of finding a minimum anti-forcing set for shortest $s$-$t$ paths, which is NP-complete. To complement this intractability result, we design fixed-parameter tractable algorithms for finding a minimum anti-forcing set for shortest $s$-$t$ paths.

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BibTeX RIS

Tatsuya Gima, Andreas Grigorjew, Yasuaki Kobayashi, Michael Lampis, Yiren Lu, Valia Mitsou, Edouard Nemery, Yuto Okada, Yota Otachi, Takumi Sato. 2025-09-29. Forcing a unique minimum spanning tree and a unique shortest path. https://arxiv.org/abs/2509.24309

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