A linear arrangement is a mapping $\pi$ from the $n$ vertices of a graph $G$ to $n$ distinct consecutive integers. Linear arrangements can be represented by drawing the vertices along a horizontal line and drawing the edges as semicircles above said line. In this setting, the length of an edge is defined as the absolute value of the difference between the positions of its two vertices in the arrangement, and the cost of an arrangement as the sum of all edge lengths. Here we study two variants of the Maximum Linear Arrangement problem (MaxLA), which consists of finding an arrangement that maximizes the cost. In the planar variant for free trees, vertices have to be arranged in such a way that there are no edge crossings. In the projective variant for rooted trees, arrangements have to be planar and the root of the tree cannot be covered by any edge. In this paper we present algorithms that are linear in time and space to 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：线性排列是从图 $G$ 的 $n$ 个顶点到 $n$ 个不同连续整数的映射 $\pi$。线性排列可通过将顶点沿水平线绘制，并将边以上方半圆弧的形式表示。在此设定中，边的长度定义为排列中其两个顶点位置之差的绝对值，排列的成本定义为所有边长的总和。本文研究了最大线性排列问题（MaxLA）的两种变体，该问题旨在寻找使成本最大化的排列。在自由树的平面变体中，顶点需以使无边交叉的方式排列。在有根树的投影变体中，排列需满足平面性，且树的根不能被任何边覆盖。本文提出了时间和空间复杂度均为线性的算法，用于解决树的平面和投影MaxLA问题。我们还证明了最大投影和平面排列的若干性质，并表明毛虫树在固定大小的所有树中使平面MaxLA最大化，从而推广了先前关于树的极值结果。