An $h$-queue layout of a graph $G$ consists of a linear order of its vertices and a partition of its edges into $h$ queues, such that no two independent edges of the same queue nest. The minimum $h$ such that $G$ admits an $h$-queue layout is the queue number of $G$. We present two fixed-parameter tractable algorithms that exploit structural properties of graphs to compute optimal queue layouts. As our first result, we show that deciding whether a graph $G$ has queue number $1$ and computing a corresponding layout is fixed-parameter tractable when parameterized by the treedepth of $G$. Our second result then uses a more restrictive parameter, the vertex cover number, to solve the problem for arbitrary $h$.

中文翻译：

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队列布局的参数化算法

图 $G$ 的 $h$ 队列布局由其顶点的线性顺序和将其边划分为 $h$ 队列组成，这样同一队列的两条独立边就不会嵌套。使 $G$ 接受 $h$-队列布局的最小 $h$ 是 $G$ 的队列号。我们提出了两种固定参数易处理算法，它们利用图的结构特性来计算最佳队列布局。作为我们的第一个结果，我们表明，当由 $G$ 的树深度参数化时，决定图 $G$ 是否具有队列号 $1$ 并计算相应的布局是固定参数易处理的。然后我们的第二个结果使用了一个更具限制性的参数，即顶点覆盖数，来解决任意 $h$ 的问题。