An SPQR-Tree-Like Embedding Representation for Upward Planarity

Brückner, Guido; Himmel, Markus; Rutter, Ignaz

doi:10.1007/978-3-030-35802-0_39

Cited by 6 publications

(5 citation statements)

References 48 publications

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“…For orthogonal drawings, the SPQR-tree played a key role in going from optimizing a given fixed embedding [43] to optimizing across all possible embeddings of a graph [7]. A recent SPQR-tree-like embedding representation for level planarity [11] might enable a similar leap for λ-drawings.…”

Section: Discussionmentioning

confidence: 99%

Level-Planar Drawings with Few Slopes

2021

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We introduce and study level-planar straight-line drawings with a fixed number $$\lambda $$ λ of slopes. For proper level graphs (all edges connect vertices of adjacent levels), we give an $$O(n \log ^2 n / \log \log n)$$ O ( n log 2 n / log log n ) -time algorithm that either finds such a drawing or determines that no such drawing exists. Moreover, we consider the partial drawing extension problem, where we seek to extend an immutable drawing of a subgraph to a drawing of the whole graph, and the simultaneous drawing problem, which asks about the existence of drawings of two graphs whose restrictions to their shared subgraph coincide. We present $$O(n^{4/3} \log n)$$ O ( n 4 / 3 log n ) -time and $$O(\lambda n^{10/3} \log n)$$ O ( λ n 10 / 3 log n ) -time algorithms for these respective problems on proper level-planar graphs. We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with $$\lambda $$ λ slopes is -hard even in restricted cases.

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Section: Discussionmentioning

confidence: 99%

Level-Planar Drawings with Few Slopes

2021

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“…For special cases, upward-planarity testing in the variable embedding setting can be performed in linear time: e.g. if the DAG has only one source [5,8,19], or if the underlying undirected graph is series-parallel [14]. Furthermore, parameterized algorithms for upward-planarity testing exist with respect to the number of sources or the treewidth of the input DAG [10].…”

Section: Introductionmentioning

confidence: 99%

On Upward-Planar L-Drawings of Graphs

Angelini¹,

Chaplick²,

Cornelsen³

et al. 2022

Preprint

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In an upward-planar L-drawing of a directed acyclic graph (DAG) each edge e is represented as a polyline composed of a vertical segment with its lowest endpoint at the tail of e and of a horizontal segment ending at the head of e. Distinct edges may overlap, but not cross. Recently, upward-planar L-drawings have been studied for st-graphs, i.e., planar DAGs with a single source s and a single sink t containing an edge directed from s to t. It is known that a plane st-graph, i.e., an embedded st-graph in which the edge (s, t) is incident to the outer face, admits an upward-planar L-drawing if and only if it admits a bitonic st-ordering, which can be tested in linear time.We study upward-planar L-drawings of DAGs that are not necessarily st-graphs. On the combinatorial side, we show that a plane DAG admits an upward-planar L-drawing if and only if it is a subgraph of a plane st-graph admitting a bitonic st-ordering. This allows us to show that not every tree with a fixed bimodal embedding admits an upward-planar L-drawing. Moreover, we prove that any acyclic cactus with a single source (or a single sink) admits an upward-planar L-drawing, which respects a given outerplanar embedding if there are no transitive edges. On the algorithmic side, we consider DAGs with a single source (or a single sink). We give linear-time testing algorithms for these DAGs in two cases: (i) when the drawing must respect a prescribed embedding and (ii) when no restriction is given on the embedding, but each biconnected component is series-parallel.

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“…While testing upward planarity is NP-complete for general graphs [15], polynomial algorithms have been devised for several classes of directed graphs. One of the most relevant such results is the linear algorithm for single-source directed graphs [7,8,22]. Other classes for which upward planarity can be tested in polynomial time are graphs with a fixed (upward) embedding [6], outerplanar graphs [27], and series-parallel graphs [11].…”

Section: Introductionmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

“…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]. For st-graphs the running time was recently improved to O(n log n) [25].…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Embeddings of Dynamic Single-Source Upward Planar Graphs

Hoog¹,

Parada²,

Rotenberg³

2022

Preprint

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A directed graph G is upward planar if it admits a planar embedding such that each edge is y-monotone. Unlike planarity testing, upward planarity testing is NPhard except in restricted cases, such as when the graph has the single-source property (i.e. each connected component only has one source).In this paper, we present a dynamic algorithm for maintaining a combinatorial embedding E(G) of a single-source upward planar graph subject to edge deletions, edge contractions, edge insertions upwards across a face, and single-source-preserving vertex splits through specified corners. We furthermore support changes to the embedding E(G) on the form of subgraph flips that mirror or slide the placement of a subgraph that is connected to the rest of the graph via at most two vertices.All update operations are supported as long as the graph remains upward planar, and all queries are supported as long as the graph remains single-source. Updates that violate upward planarity are identified as such and rejected by our update algorithm. We dynamically maintain a linear-size data structure on G which supports incidence queries between a vertex and a face, and upward-linkability of vertex pairs. If a pair of vertices are not upwards-linkable, we facilitate one-flip-linkable queries that point to a subgraph flip that makes them linkable, if any such flip exists.We support all updates and queries in O(log 2 n) time.

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An SPQR-Tree-Like Embedding Representation for Upward Planarity

Cited by 6 publications

References 48 publications

Level-Planar Drawings with Few Slopes

Level-Planar Drawings with Few Slopes

On Upward-Planar L-Drawings of Graphs

Dynamic Embeddings of Dynamic Single-Source Upward Planar Graphs

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