Given a graph $G$, a set $T$ of terminal vertices, and a demand graph $H$ on $T$, the \textsc{Multicut} problem asks for a set of edges of minimum weight that separates the pairs of terminals specified by the edges of $H$. The \textsc{Multicut} problem can be solved in polynomial time if the number of terminals and the genus of the graph is bounded (Colin de Verdière [Algorithmica, 2017]). Focke et al.~[SoCG 2024] characterized which special cases of Multicut are fixed-parameter tractable parameterized by the number of terminals on planar graphs. Moreover, they precisely determined how the parameter genus influences the complexity and presented partial results of this form for graphs that can be made planar by the deletion of $π$ edges. We complete the picture on how this parameter $π$ influences the complexity of different special cases and precisely determine the influence of the crossing number. Formally, let $\mathcal{H}$ be any class of graphs (satisfying a mild closure property) and let Multicut$(\mathcal{H})$ be the special case when the demand graph $H$ is in $\mathcal{H}$. Our first main result is showing that if $\mathcal{H}$ has the combinatorial property of having bounded distance to extended bicliques, then Multicut$(\mathcal{H})$ on unweighted graphs is FPT parameterized by the number $t$ of terminals and $π$. For the case when $\mathcal{H}$ does not have this combinatorial property, Focke et al.~[SoCG 2024] showed that $O(\sqrt{t})$ is essentially the best possible exponent of the running time; together with our result, this gives a complete understanding of how the parameter $π$ influences complexity on unweighted graphs. Our second main result is giving an algorithm whose existence shows that the parameter crossing number behaves analogously if we consider weighted graphs.

翻译：给定图$G$、终端顶点集合$T$以及$T$上的需求图$H$，\textsc{多割}问题要求找到一组权重最小的边，以分离由$H$的边所指定的终端对。若终端数量与图的亏格有界，则\textsc{多割}问题可在多项式时间内求解（Colin de Verdière [Algorithmica, 2017]）。Focke等人~[SoCG 2024]刻画了多割问题在平面图上哪些特例关于终端数量是固定参数可解的。此外，他们精确确定了参数亏格如何影响复杂度，并针对可通过删除$π$条边变为平面的图给出了此类形式的局部结果。我们完整揭示了参数$π$如何影响不同特例的复杂度，并精确确定了交叉数的影响。形式化地，令$\mathcal{H}$为任意图类（满足温和闭包性质），并令Multicut$(\mathcal{H})$表示需求图$H$属于$\mathcal{H}$的特例。我们的第一个主要结果表明，若$\mathcal{H}$具有组合性质——即到扩展双分团的有界距离，则无权图上的Multicut$(\mathcal{H})$关于终端数量$t$与参数$π$是固定参数可解的。对于$\mathcal{H}$不具备此组合性质的情形，Focke等人~[SoCG 2024]证明$O(\sqrt{t})$本质上是运行时间指数的最佳可能上界；结合我们的结果，这给出了参数$π$在无权图上如何影响复杂度的完整理解。我们的第二个主要结果是给出一个算法，其存在性表明若考虑带权图，参数交叉数具有类似的行为特性。