A Sublinear Bound on the Page Number of Upward Planar Graphs

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 2/3 (n)).

Directed Acyclic Outerplanar Graphs Have Constant Stack Number

“…(E.g., Figure 2 depicts such an example.) Superseding previous results [21], Jungeblut, Merker and Ueckerdt [34] recently gave the first sublinear upper bound for all upward planar graphs by showing that every n-vertex upward planar graph G has stack number sn(G)…”

“…A slight generalization, namely whether all upward planar graphs have bounded stack number, is considered to be one of the most important open questions in the field of linear layouts [21,41,34]. It is known to hold for upward planar 3-trees [21,41].…”

“…In fact, this is just a special case of the same question for general upward planar graphs. Here the best lower bound is an upward planar graph requiring five stacks compared to an O((n log n) 2/3 ) upper bound, where n is the number of vertices [34].…”

Recognizing DAGs with Page-Number 2 Is NP-complete

“…Heath and Pemmaraju [16] showed that there exist planar DAGs whose pagenumber is linear in the input size. Certain subfamilies of planar DAGs, however, have bounded page-number [1,6,10,18], while recently it was shown that upward planar graphs have sublinear page-number [20], improving previous bounds [12]. From an algorithmic point of view, testing whether a DAG has page-number k is NP-complete for every fixed value of k ≥ 3 [7], linear-time solvable for k = 1 [16], and fixed-parameter tractable with respect to the vertex cover number for every k [6] and with respect to the treewidth for st-graphs when k = 2 [7].…”

Dynamic Embeddings of Dynamic Single-Source Upward Planar Graphs

“…There are no existing worst-case results for dynamic upward embeddings and no results for dynamic upward embeddings subject to flips in the embedding. The study of upward planar graphs continues to be a prolific area of research, including recent developments in parameterized algorithms for upward planarity [10], bounds on the page number [23], morphing [26], and extension questions [8,25]. In particular, it can be tested in O(n 2 ) time whether a given drawing can be extended to an upward planar drawing of an n-vertex single-source directed graph G [8].…”