We present proof labeling schemes for graphs with bounded pathwidth that can decide any graph property expressible in monadic second-order (MSO) logic using $O(\log n)$-bit vertex labels. Examples of such properties include planarity, Hamiltonicity, $k$-colorability, $H$-minor-freeness, admitting a perfect matching, and having a vertex cover of a given size. Our proof labeling schemes improve upon a recent result by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), which achieved the same result for graphs of bounded treewidth but required $O(\log^2 n)$-bit labels. Our improved label size $O(\log n)$ is optimal, as it is well-known that any proof labeling scheme that accepts paths and rejects cycles requires labels of size $\Omega(\log n)$. Our result implies that graphs with pathwidth at most $k$ can be certified using $O(\log n)$-bit labels for any fixed constant $k$. Applying the Excluding Forest Theorem of Robertson and Seymour, we deduce that the class of $F$-minor-free graphs can be certified with $O(\log n)$-bit labels for any fixed forest $F$, thereby providing an affirmative answer to an open question posed by Bousquet, Feuilloley, and Pierron (Journal of Parallel and Distributed Computing 2024).

翻译：我们为有界路径宽度的图提出了证明标记方案，该方案能够使用$O(\log n)$比特的顶点标签判定任何可用一元二阶（MSO）逻辑表达的图性质。此类性质的例子包括平面性、哈密顿性、$k$-可着色性、$H$-子式自由性、允许完美匹配以及具有给定大小的顶点覆盖。我们的证明标记方案改进了Fraigniaud、Montealegre、Rapaport和Todinca（Algorithmica 2024）最近的结果，该结果对有界树宽图实现了相同的结论，但需要$O(\log^2 n)$比特的标签。我们改进后的标签大小$O(\log n)$是最优的，因为众所周知，任何接受路径但拒绝环路的证明标记方案都需要$\Omega(\log n)$比特大小的标签。我们的结果表明，对于任何固定常数$k$，路径宽度至多为$k$的图可以使用$O(\log n)$比特的标签进行验证。应用Robertson和Seymour的排除森林定理，我们推导出对于任何固定森林$F$，$F$-子式自由图类可以用$O(\log n)$比特的标签进行验证，从而为Bousquet、Feuilloley和Pierron（Journal of Parallel and Distributed Computing 2024）提出的一个开放问题提供了肯定答案。