A Sublinear Bound on the Page Number of Upward Planar Graphs

Paul Jungeblut,Laura Merker,Torsten Ueckerdt

doi:10.1137/1.9781611977073.42

Abstract

The page number of a directed acyclic graph $G$ is the minimum $k$ for which there is a topological ordering of $G$ and a $k$-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We address the long-standing open problem asking for the largest page number among all upward planar graphs. We improve the best known lower bound to $5$ and present the first asymptotic improvement over the trivial $O(n)$ upper bound, where $n$ denotes the number of vertices in $G$. Specifically, we first prove that the page number of every upward planar graph is bounded in terms of its width, as well as its height. We then combine both approaches to show that every $n$-vertex upward planar graph has page number $O(n^{2/3} \log(n)^{2/3})$.

Highlights

In an upward planar drawing of a directed acyclic graph G = (V, E), every vertex v ∈ V is a point in the Euclidean plane, and every edge (u, v) ∈ E is a strictly y-monotone curve1 with lower endpoint u and upper endpoint v that is disjoint from other points and curves, except in its endpoints
In a book embedding of a directed acyclic graph G = (V, E), the vertex set V is endowed with a topological ordering
We remark that with a very recent improvement of Observation 1.1 by Davies [12], we obtain O(h log(h)) as an upper bound on the page number of upward planar graphs with height h

Summary

Introduction

Book embeddings of directed graphs were first considered by Nowakowski and Parker [28] in 1989 They introduced the page number of a poset P by considering its cover graph G(P ) and restricting the spine ordering to be a topological ordering of G(P ), or equivalently, a linear extension of P. We bound the page number of upward planar graphs G in terms of their width. We remark that with a very recent (and yet unpublished) improvement of Observation 1.1 by Davies [12], we obtain O(h log(h)) as an upper bound on the page number of upward planar graphs with height h. We improve the best known lower bound on the maximum twist number and page number among upward planar graphs to 5. There is an upward planar graph G with pn(G) tn(G) 5

Bounded Width

Bounded Height

Bound in Terms of the Number of Vertices

Lower Bound

Conclusions