Kolja Kühn | Searcharxiv

arXiv2026

Completing the Complexity Classification of 2-Solo Chess: Knights and Kings are Hard

We extend the study of the 2-Solo Chess problem which was first introduced by Aravind, Misra, and Mittal in 2022. 2-Solo Chess is a single-player variant of chess in which the player must clear the board via captures such that only one piece remains, with each piece capturing at most twice. It is known that the problem is solvable in polynomial time for instances containing only pawns, while it becomes NP-complete for instances restricted to rooks, bishops, or queens. In this work, we complete the complexity classification by proving that 2-Solo Chess is NP-complete if the instance contains only knights or only kings.

cs.CC↗

arXiv2025

Computing Treedepth Obstructions

The graph parameter treedepth is minor-monotone; hence, the class of graphs with treedepth at most $k$ is minor-closed. By the Graph Minor Theorem, such a class is characterized by a finite set of forbidden minors. A conjecture of Dvořák, Giannopoulou, and Thilikos states that every such forbidden minor has at most $2^k$ vertices. We present an algorithm that, given $n, k \in \mathbb{N}$, computes the set of forbidden minors, forbidden subgraphs, and forbidden induced subgraphs on at most $n$ vertices, for the class of graphs of treedepth at most $k$. Applying this algorithm to $k = 4$ and $n = 16$, we enumerate 1546 forbidden minors, 1718 forbidden subgraphs, and 12204 forbidden induced subgraphs. Assuming the above conjecture holds, these sets constitute the complete obstruction sets for graphs of treedepth at most 4.

cs.DM↗