AI 中文总结
本研究针对具有积结构的图，提出算法计算轨道布局，给出轨道数上界，为平面图等多类图建立新轨道数上界，算法运行高效且仅用基础链表数据结构。
AI 中文摘要
图的轨道布局是将其顶点划分为线性有序的独立集（称为轨道），使得同一对轨道之间的任意两条边不交叉。给定一个图，该场景下的目标是确定其轨道数，即使该图能容纳轨道布局所需的最少轨道数。本研究针对具有积结构的图，给出了轨道数的上界。我们的主要贡献是提出一种算法，对于强积 $P^h \boxtimes K_r \boxtimes H$ 的每个子图，该算法可计算出轨道数不超过 $(2h+1) \cdot r \cdot tn(H)$ 的轨道布局，其中 $P^h$ 是路径 $P$ 的 $h$ 次幂，$K_r$ 是 $r$ 个顶点的完全图，$H$ 是轨道数为 $tn(H)$ 的图。结合文献中现有的积结构结果，该算法为若干图类提供了轨道数的上界：对于平面图，所得上界与当前最优的225一致；对于1-平面图和最优2-平面图，该算法得到的轨道布局最多使用375条轨道；对于亏格为$k$的图、$k$-平面图、$k$-框架图、$k$-地图图和$k$-字符串图，该算法提供的轨道布局所需轨道数仅依赖于$k$，从而为这些图类建立了新的轨道数上界。该算法对平面图的运行时间为线性时间，更一般地，当输入的$n$顶点图附带对应的积结构分解时，算法运行时间为$O(n + h \cdot r \cdot t + f_t(H))$，其中$t=tn(H)$，$f_t(H)$是计算$H$的$t$轨道布局所需的时间。此外，该算法仅使用基础链表数据结构。
英文摘要
A track layout of a graph is a partition of its vertices into linearly ordered independent sets, called tracks, such that no two edges between the same pair of tracks cross. Given a graph, the goal in this context is to determine its track number, that is, the minimum number of tracks required for the graph to admit a track layout. In this work, we present upper bounds on the track number of graphs admitting a product structure. Our main contribution is an algorithm that computes a track layout with at most $(2h+1) \cdot r \cdot tn(H)$ tracks for every subgraph of the strong product $P^h \boxtimes K_r \boxtimes H$, where $P^h$ is the $h$-th power of a path $P$, $K_r$ is the complete graph on $r$ vertices, and $H$ is a graph with track number $tn(H)$. Combined with existing product-structure results from the literature, this algorithm yields upper bounds on the track number of several graph classes. For planar graphs, the obtained bound matches the current best-known upper bound of $225$. For $1$-planar and optimal $2$-planar graphs, our algorithm yields track layouts with at most $375$ tracks, while for genus-$k$, $k$-planar, $k$-framed, $k$-map, and $k$-string graphs it provides track layouts with a number of tracks that depends solely on $k$, thus establishing new upper bounds on the track number for these graph classes. The algorithm runs in linear time for planar graphs and, more generally, in $O(n + h \cdot r \cdot t + f_t(H))$ time whenever a corresponding product-structure decomposition of the input $n$-vertex graph is provided as part of the input, where $t=tn(H)$ and $f_t(H)$ is the time needed to compute a $t$-track layout of $H$. Furthermore, our algorithm only uses elementary linked-list data structures.
