On Planar Greedy Drawings of 3-Connected Planar Graphs

Lozzo, Giordano Da; D’Angelo, Anthony; Frati, Fabrizio

doi:10.1007/s00454-018-0001-5

Cited by 7 publications

(3 citation statements)

References 38 publications

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“…The rich literature on greedy drawings described above witnesses the relevance of these kind of drawings both from a practical and from a theoretical perspective. In particular, our work enhances the research on greedy drawings that satisfy some interesting topological or geometric requirements, such as planarity [8] and face convexity [17,23,29].…”

Section: Introductionmentioning

confidence: 81%

Greedy Rectilinear Drawings

Angelini

Bekos

Didimo

et al. 2018

Lecture Notes in Computer Science

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A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. From an application perspective, greedy drawings are especially relevant to support routing schemes in ad hoc wireless networks. The existence of greedy drawings has been widely studied under different topological and geometric constraints, such as planarity, face convexity, and drawing succinctness. We introduce greedy rectilinear drawings, where edges are horizontal or vertical segments. These drawings have several properties that improve human readability and support network routing.We address the problem of testing whether a planar rectilinear representation, i.e., a plane graph with prescribed vertex angles, admits a greedy rectilinear drawing. We give a characterization, a lineartime testing algorithm, and a full generative scheme for universal greedy rectilinear representations, i.e., those for which every drawing is greedy. For general greedy rectilinear representations, we give a combinatorial characterization and, based on it, a polynomial-time testing and drawing algorithm for a meaningful subset of instances.

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Section: Introductionmentioning

confidence: 81%

Greedy Rectilinear Drawings

Angelini

Bekos

Didimo

et al. 2018

Lecture Notes in Computer Science

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“…The conjecture itself was proved by Leighton and Moitra [8] and Angelini et al [2] independently. Later, Da Lozzo et al [4] proved a stronger version of the conjecture, which claims existence of a planar greedy drawing. As greedy-drawability is a monotonic graph property (i.e., adding an edge preserves greedy-drawability), the class of trees has also gained much attention.…”

Section: Introductionmentioning

confidence: 99%

Complete combinatorial characterization of greedy-drawable trees

Miyata¹,

Nosaka²

2022

Preprint

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A (Euclidean) greedy drawing of a graph is a drawing in which, for any two vertices s, t (s = t), there is a neighbor vertex of s that is closer to t than to s in the Euclidean distance. Greedy drawings are important in the context of message routing in networks, and graph classes that admit greedy drawings have been actively studied. Nöllenburg and Prutkin (Discrete Comput. Geom., 58(3), pp. 543-579, 2017) gave a characterization of greedy-drawable trees in terms of an inequality system that contains a non-linear equation. Using the characterization, they gave a linear-time recognition algorithm for greedy-drawable trees of maximum degree ≤ 4. However, a combinatorial characterization of greedy-drawable trees of maximum degree 5 was left open. In this paper, we give a combinatorial characterization of greedy-drawable trees of maximum degree 5, which leads to a complete combinatorial characterization of greedy-drawable trees.Furthermore, we give a characterization of greedy-drawable pseudo-trees. It turns out that greedydrawable pseudo-trees admit a simpler characterization than greedy-drawable trees.

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“…Graph embeddings in which every pair of vertices is connected by a path satisfying certain geometric properties have been the subject of intensive research. As the most notorious example, a greedy drawing of a graph [5,7,19,24,31,36,40,42,43,49] is such that, for every pair of vertices u and v, there is a path from u to v that monotonically decreases the distance to v at every vertex. More restricted than greedy drawings are self-approaching and increasing-chord drawings [3,20,41].…”

Section: Introductionmentioning

confidence: 99%

Drawing Graphs as Spanners

Aichholzer¹,

Borrazzo²,

Bose³

et al. 2020

Preprint

Self Cite

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We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph G, the goal is to construct a straight-line drawing Γ of G in the plane such that, for any two vertices u and v of G, the ratio between the minimum length of any path from u to v and the Euclidean distance between u and v is small. The maximum such ratio, over all pairs of vertices of G, is the spanning ratio of Γ . First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio 1, a proper straight-line drawing with spanning ratio 1, and a planar straight-line drawing with spanning ratio 1 are NP-complete, ∃R-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio 1 to spanning ratio 1 + allows us to draw every graph. Namely, we prove that, for every > 0, every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than 1 + . Third, our drawings with spanning ratio smaller than 1 + have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.

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On Planar Greedy Drawings of 3-Connected Planar Graphs

Cited by 7 publications

References 38 publications

Greedy Rectilinear Drawings

Greedy Rectilinear Drawings

Complete combinatorial characterization of greedy-drawable trees

Drawing Graphs as Spanners

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