The maximum linear arrangement problem for trees under projectivity and planarity
A linear arrangement is a mapping p 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 t...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/388702 |
| Acceso en línea: | https://hdl.handle.net/2117/388702 https://dx.doi.org/10.1016/j.ipl.2023.106400 |
| Access Level: | acceso abierto |
| Palabra clave: | Trees (Graph theory) Graph algorithms Linear arrangements Maximum linear arrangement problem Projectivity Planarity Arbres (Teoria de grafs) Àrees temàtiques de la UPC::Informàtica::Informàtica teòrica::Algorísmica i teoria de la complexitat |
| Sumario: | A linear arrangement is a mapping p 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. |
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