The Maximum Linear Arrangement Problem for trees under projectivity and planarity

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.