AI 中文总结
研究图的加权书本厚度，它是边出现的最小可实现平均页数。证明了一些平面图和\(2 -\)树的加权书本厚度实现情况，给出路径宽度至多为\(2\)的图的相关结论，还表明判定加权书本厚度是否至多为\(k\)是NP完全问题。
AI 中文摘要
我们引入并研究了图的加权书本厚度。图\(G=(V,E)\)的\(k\)页书本嵌入由\(V\)的生成圈\(C\)（不一定是\(G\)的一部分）和划分\(E=\bigcup_{i=1}^{k}E_i\)定义，使得\(E\cap C\subseteq E_1\)，且每个图\(G_i=(V,E_i\cup C)\)（\(1 \le i \le k\)）以外圈\(C\)为外平面图。若\(e\in E_i\)，则称\(e\)出现在第\(i\)页。图\(G\)的经典书本厚度是使得存在\(k\)页书本嵌入的最小\(k\)，而加权书本厚度是边出现的最小可实现平均页数。实现加权书本厚度的嵌入可能与实现经典书本厚度的不同。我们证明，最多九个顶点的平面图都有一个实现其加权书本厚度的\(2\)页书本嵌入，但十个顶点的平面图中，存在一个平面图，其加权书本厚度的每个实现所需页数比其书本厚度多。还证明存在一个\(2 -\)树，其加权书本厚度不能在两页上实现。另一方面，我们表明对于路径宽度至多为\(2\)的每个图，加权书本厚度总能通过\(2\)页书本嵌入实现，且可在线性时间内找到这样的嵌入。此外，我们证明对于给定整数\(k\)，判定加权书本厚度是否至多为\(k\)是NP完全问题。
英文摘要
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 $k$.
CommentsAppears in the Proceedings of the 34th International Symposium on Graph Drawing and Network Visualization (GD 2026)
