The disjoint paths logic, FOL+DP, is an extension of First-Order Logic (FOL) with the extra atomic predicate ${\sf dp}_k(x_1,y_1,\ldots,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in\{1,\ldots, k\}$. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every proper minor-closed graph class, model-checking for FOL+DP can be done in quadratic time. We also introduce an extension of FOL+DP, namely the scattered disjoint paths logic, FOL+SDP, where we further consider the atomic predicate $s{\sf -sdp}_k(x_1,y_1,\ldots,x_k,y_k),$ demanding that the disjoint paths are within distance bigger than some fixed value $s$. Using the same technique we prove that model-checking for FOL+SDP can be done in quadratic time on classes of graphs with bounded Euler genus.
翻译:脱节路径逻辑 FOL+DP 是第一正弦逻辑( FOL) 的延伸 。 此逻辑可以表达远端逻辑( FOL) 的多种问题, 无法显示 FOL 的表达潜力 。 我们证明, 对于每一个合适的小封闭图形类, FOL+DP 的模型检查可以在四边形时间进行。 我们还引入了 FOL+DP 的扩展, 即分散的脱节路径逻辑, FOL+SDP, 我们在此进一步考虑 $sf - sdpäk (x_ 1,y_ 1,\ldots,x_k,y_k,y_k) 的原子上游路径 $ $sfsf - sdpäk (x_ 1,\ldots,x_k,y_k), 要求断绝路径在距离大于某些固定值 $。 我们用同样的技术来证明, 与 OL+ DP 绑定的 度的 矩形 的 矩形 将 校验为 FOL+ SDP 。