Disjoint-paths logic, denoted $\mathsf{FO}$+$\mathsf{DP}$, extends first-order logic ($\mathsf{FO}$) with atomic predicates $\mathsf{dp}_k[(x_1,y_1),\ldots,(x_k,y_k)]$, expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i$, for $1\leq i\leq k$. We prove that for every graph class excluding some fixed graph as a topological minor, the model checking problem for $\mathsf{FO}$+$\mathsf{DP}$ is fixed-parameter tractable. This essentially settles the question of tractable model checking for this logic on subgraph-closed classes, since the problem is hard on subgraph-closed classes not excluding a topological minor (assuming a further mild condition of efficiency of encoding).
翻译:disjoint- paths 逻辑, 表示$\ mathsf{FO} $+$\ mathsf{DP}, 延伸第一阶逻辑 (\mathsf{FO}$), 延伸第一阶逻辑 (\mathsf{dp}$), 延伸第一阶逻辑 (\mathsf{dp{k}$), 延伸至原子上游 $\mathsf{dp} k[(x_ 1,y_ 1,\ldlots,\ldots, (x_k,y_k)] 美元, 表示$x_ i美元和$y_ i 美元之间存在内部的顶端点断路径。 我们证明, 对于每个图表类别中不包括某些固定的表层小图, 都存在固定为可移动参数的模型检查问题。 这基本上解决了子图表封闭类中此逻辑的可移动模型检查问题, 因为问题在子闭类中很难排除一个表层小小( 假设编码效率的更温)。