The Maximum Linear Arrangement problem (MaxLA) consists of finding a mapping $\pi$ from the $n$ vertices of a graph $G$ to distinct consecutive integers that maximizes $D(G)=\sum_{uv\in E(G)}|\pi(u) - \pi(v)|$. In this setting, vertices are considered to lie on a horizontal line and edges are drawn as semicircles above the line. There exist variants of MaxLA in which the arrangements are constrained. In the planar variant, edge crossings are forbidden. In the projective variant for rooted trees, arrangements are planar and the root cannot be covered by any edge. Here we present $O(n)$-time and $O(n)$-space algorithms that solve planar and projective MaxLA for trees. We also prove several properties of maximum projective and planar arrangements, and show that caterpillar trees maximize planar MaxLA over all trees of a fixed size thereby generalizing a previous extremal result on trees.

翻译：最大线性排列问题（MaxLA）旨在寻找一个映射 $\pi$，将图 $G$ 的 $n$ 个顶点映射至互不相同的连续整数，使得 $D(G)=\sum_{uv\in E(G)}|\pi(u) - \pi(v)|$ 最大化。在此设定中，顶点被视为位于水平线上，边则以上方半圆弧线绘制。MaxLA存在若干受约束的变体：在平面性变体中，禁止边交叉；在有根树的投影性变体中，排列需满足平面性且根节点不能被任何边覆盖。本文提出了时间复杂度为 $O(n)$、空间复杂度为 $O(n)$ 的算法，用于求解树的平面性与投影性MaxLA问题。此外，我们证明了最大投影性排列与平面性排列的若干性质，并揭示了毛虫树在固定规模的所有树中最大化平面性MaxLA值，从而推广了此前关于树的极值结论。