Weighted Book Thickness

2026-07-27

AI Summary

This work proposes and systematically investigates the weighted book thickness of graphs, which seeks to minimize the average page number assigned to edges rather than the maximum page number as in classical book thickness. By constructing a vertex ordering and partitioning edges such that each page together with the ordering induces an outerplanar graph, the study integrates techniques from graph embeddings, outerplanar graph theory, pathwidth analysis, and computational complexity. The main contributions include the first formal definition of weighted book thickness and a demonstration of its fundamental distinction from classical book thickness; a proof that graphs of pathwidth at most two admit an optimal two-page weighted book embedding; a ten-vertex planar counterexample showing that some 2-trees cannot achieve optimality with only two pages; and the establishment that deciding whether the weighted book thickness is at most $k$ is NP-complete.

Abstract

We introduce and study the weighted book thickness of graphs. A $k$-page book embedding of a graph $G=(V,E)$ is defined by a spanning cycle $C$ for $V$ (which does not need to be part of $G$) and a partition $E=\bigcup_{i=1}^{k}E_i$ such that $E\cap C\subseteq E_1$ and each graph $G_i=(V,E_i\cup C)$, for $1 \le i \le k$, is outerplane with outer cycle $C$. If $e\in E_i$, we say that $e$ appears on Page $i$. The classical book thickness of a graph $G$ is the minimum $k$ such that there exists a $k$-page book embedding of $G$, that is, the minimum (over all book embeddings of $G$) achievable maximum page an edge appears on. In contrast, the weighted book thickness is the minimum achievable average page an edge appears on. The embeddings that realize weighted book thickness can differ from those that realize (classical) book thickness. We show that, although every planar graph on at most nine vertices admits a 2-page book embedding realizing its weighted book thickness, already for ten vertices, there is a planar graph for which every realization of its weighted book thickness needs more pages than its book thickness. We prove that there even exists a 2-tree whose weighted book thickness cannot be realized on two pages. On the positive side, we show that for every graph of pathwidth at most two, the weighted book thickness can always be realized by a 2-page book embedding and such an embedding can be found in linear time. Moreover, we prove that it is NP-complete to decide if the weighted book thickness is at most $k$, for some given integer $k$.

Problem

• weighted book thickness
• book embedding
• graph drawing
• planar graph
• NP-completeness

Innovation

• weighted book thickness
• book embedding
• outerplanar graph
• pathwidth
• NP-completeness