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：最大线性布局问题（MaxLA）包括寻找从图$G$的$n$个顶点到不同连续整数的映射$\pi$，使得$D(G)=\sum_{uv\in E(G)}|\pi(u)-\pi(v)|$最大化。在此设置中，顶点被认为位于水平线上，边被绘制为在线上方的半圆形。存在MaxLA的各种变体，在其中约束了排列形式。在平面变体中，禁止边交叉。在针对有根树的投影变体中，排列是平面的，根不能被任何边覆盖。在本文中，我们提供了解决树的平面和投影MaxLA的$O(n)$时间和$O(n)$空间算法。我们还证明了最大投影和平面排列的几个属性，并表明毛毛虫树在所有固定大小的树中最大化平面MaxLA，从而推广了先前的极端结果。