Optimal Morphs of Planar Orthogonal Drawings II

Arthur van Goethem; Bettina Speckmann; Kevin Verbeek

Optimal Morphs of Planar Orthogonal Drawings II

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Pith's one-line read This paper proves that equivalent planar orthogonal drawings of a disconnected graph morph into each other with at most $s+1$ linear morphs, where $s$ is the spirality of the starting drawing and $s = O(n)$.

desk verdict A solid, genuinely new result that closes the disconnected orthogonal morphing problem at the asymptotically optimal O(n) bound, though it leans heavily on the authors' prior framework. read the letter →

arxiv 1908.08365 v1 pith:VHQBYQWL submitted 2019-08-22 cs.CG

Arthur van Goethem , Bettina Speckmann , Kevin Verbeek This is my paper · ORCID

classification cs.CG

keywords planarorthogonaldrawingsmorphingwiresspiralitydisconnectedgraphslinearmorphsgraphdrawingplanarity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Two planar orthogonal drawings of the same graph can be reshaped into each other by a sequence of simple linear interpolations without ever losing planarity, orthogonality, or reasonable drawing complexity. This paper proves that the number of such linear morphs is at most $s+1$, where $s$ is the spirality of the starting drawing relative to the target; since a suitable wire set always has $s = O(n)$, the bound is $O(n)$ even when the graph is disconnected. Earlier methods needed $O(n^{1.5})$ morphs in the disconnected case, so this matches the lower bound that was already known for straight-line drawings. The proof works by picking separator curves (wires) of low spirality, then compressing each unit drop of spirality into one linear morph, with a rerouting and redrawing step that keeps intermediate complexity linear at the cost of one extra morph.

What carries the argument

The central object is a wire, an orthogonal polyline that separates the drawing into left/right or top/bottom parts and records the order in which vertices, edges, and other wires are crossed; a set of matched left-to-right and top-to-bottom wires encodes the structural difference between the two drawings. Spirality of a wire is the maximum absolute number of left turns minus right turns accumulated along any of its links, and it serves as the distance measure being driven down to zero. The argument is carried by three mechanisms: shortest-compatible wire insertion, which pins spirality to $O(n)$; staircases, which are chains of maximum-spirality links that isolate the pairs of vertices whose order changes, enabling each unit of spirality to be removed by one linear morph; and windmill rerouting with redraw steps, which keep the number of bends added per iteration linear so the final drawing has linear complexity.

What would settle it

A concrete disproof would be an explicit pair of equivalent planar orthogonal drawings of a disconnected graph with $n$ vertices for which every equivalent set of wires has spirality growing faster than linearly in $n$; such an example would refute the $O(n)$ morph theorem immediately. A narrower test is to search for a single wire whose spiral contains a layer empty of both vertices and edges but full of previously selected shortest wires; if shortcutting that layer yields a wire that is not equivalent to $\Gamma_O$, then the proof of Lemma 3 fails at that point.

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Extended reading notes

Core claim

The central claim is that two equivalent planar orthogonal drawings $\Gamma_I$ and $\Gamma_O$ of a possibly disconnected graph $G$ admit a morph into each other using $s+1$ linear morphs, where $s$ is the spirality of $\Gamma_I$ relative to $\Gamma_O$, and every intermediate drawing stays planar, orthogonal, and of linear complexity. Because Lemma 3 guarantees an equivalent set of wires with spirality $O(n)$, this gives an optimal $O(n)$ bound, matching the straight-line lower bound. The construction is connectivity-agnostic: the graph may fall into any number of components, and the wire set still has linear spirality because each wire is chosen as the shortest one compatible with its predecessors, so any overly spiral layer must contain a vertex or an edge and there can be only $O(n)$ such layers. The remaining work shows that all intermediate morphs that reduce spirality by one can be executed together in a single linear morph, and that rerouting and simplification can be adapted to this batching so intermediate complexity never grows beyond linear.

Load-bearing premise

The whole construction hangs on Lemma 3's claim that one can always choose an equivalent set of wires whose spirality is linear in the size of the drawing; that claim relies on being able to shortcut any spiral layer that contains no vertex or edge, because a layer containing only other wires would mean the chosen wire was not already the shortest compatible one. If shortcutting an empty layer ever produces a wire that is no longer equivalent to the target drawing, the linear spirality bound, and the $O(n)$ morph bound along with it, breaks.

Editorial extensions

If this is right

The $O(n)$ upper bound now holds for all planar orthogonal drawings, connected or not, and therefore matches the straight-line lower bound of $\Omega(n)$, so the worst-case number of linear morphs is optimal.
Without the rerouting-and-simplification refinement, exactly $s$ linear morphs suffice while preserving planarity and orthogonality, at the cost of $O(n^3)$ intermediate complexity; with the refinement, $s+1$ morphs keep intermediate complexity linear.
The wire construction and batching arguments do not depend on the graph being connected, so the same algorithm covers arbitrary numbers of components without any preprocessing that inflates the drawing.
Every intermediate drawing during the morph is planar, orthogonal, and of linear complexity, which means the morph can be used in interactive settings where the mental map of the drawing must be preserved.
Because spirality can be computed and the morph sequence is explicit, the result yields a constructive algorithm that runs from any two equivalent orthogonal drawings to a sequence of linear morphs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the shortest-wire argument is as robust as it appears, the same selection principle might yield low-complexity morphs for other equivalence notions—for example, orthogonal drawings with prescribed edge routes or drawings on surfaces—where separators can be described by homotopy classes rather than straight lines.
The paper leaves open whether the final morph count can be lowered from $s+1$ to exactly $s$ while still keeping all intermediate drawings of linear complexity; the tradeoff between one extra morph and quadratic-or-worse intermediate complexity is an artifact of the construction, not a proven barrier.
The windmill-rerouting technique, which bundles all wires crossing an edge into a spiral structure, may be independently useful for reducing bend complexity in other orthogonal drawing algorithms that update drawings locally.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper finishes the job the authors started in their SoCG 2018 paper: it brings disconnected planar orthogonal drawings down to O(n) linear morphs, matching the straight-line lower bound. The new exportable piece is the s+1 bound and, more importantly, the wire-construction lemma (Lemma 3) that keeps spirality O(n) even when connectivity is absent. That is a real advance, not just a tweak: the earlier connectivity augmentation route gave O(n^1.5), and the homotopy-class ambiguity for disconnected graphs is a genuine obstruction they had to engineer around.

What the paper does well: the batching argument (Theorem 2) and the rerouting/simplification lemmas are substantial, and the appendix gives real proofs rather than hand-waving. I followed the central chain from Lemma 3 to Theorem 3 and did not find a load-bearing gap. I also checked the stress-test note on Lemma 2's shortcut argument, which is the least formal step. It holds: the layer-boundary wire is strictly shortened, contradicting its choice as shortest, and the rerouting preserves the crossing order because the layer contains no vertices or edges. The terse part is the reconnection order at the gates; the appendix sketches it, and I believe it is sound, but this is where I would want a referee to spend time.

Soft spots, in proportion. First, the paper is not self-contained: it repeatedly invokes Theorem 8 and Lemma 9 from [11] as black boxes, so the reader must have that paper open. That is normal for a sequel, but worth stating. Second, the "optimal" in the title means asymptotically optimal in n; the s+1 bound itself is not shown tight, and the lower bound is inherited from the straight-line case rather than for the orthogonal-disconnected setting. Not a flaw, just calibration. Third, some proof sketches in the main text (Lemma 5, Lemma 9) are quite compressed, though the appendix expands them reasonably.

Who this is for: anyone working in graph drawing, especially orthogonal morphing and spirality. It is a meaningful closing of an asymptotic gap, not a reshaping of the field. The citation pattern is honest: the heavy self-citation is legitimate because the prior framework is being refined, and the new claims are proved from wire combinatorics rather than assumed.

I would send this to a serious referee. My own verdict is accept; I would ask the referee to check Lemma 2 and the rerouting invariance in Lemma 10 carefully, but I do not expect those to break.

Referee Report

0 major / 5 minor

Summary. This paper studies morphing between two equivalent planar orthogonal drawings ΓI and ΓO of a possibly disconnected graph G, while maintaining planarity, orthogonality, and linear complexity of intermediate drawings. Building on the connected-graph result of Van Goethem and Verbeek [11], which uses O(n) linear morphs via the spirality s of ΓI relative to ΓO, the paper proves that s+1 linear morphs always suffice, even for disconnected graphs, and since s=O(n) this is asymptotically optimal. The proof proceeds in three stages: (1) constructing an equivalent set of lr- and tb-wires with spirality O(n) for disconnected graphs (Section 3, Lemmas 1-3), (2) showing that each unit of spirality reduction can be batched into a single linear morph (Section 4, Theorem 2 and Lemmas 4-9), and (3) adapting the rerouting and simplification operations of [11] so that the batched morphs maintain linear complexity of the intermediate drawings (Section 5, Theorem 3). The paper is a full version with appendix proofs, and includes an implementation video.

Significance. If correct, this result settles the optimal linear bound for morphing planar orthogonal drawings of disconnected graphs, improving the previous O(n^1.5) bound for the disconnected case and matching the Ω(n) lower bound of Alamdari et al. for straight-line morphs. The main technical novelties are the wire-construction lemma for disconnected graphs (Lemma 3), which handles multiple homotopy classes per wire, and the batching argument (Section 4) that compresses an entire spirality unit into one linear morph. The paper is parameter-free in the sense that the construction uses no fitted constants, the lower bound is cited from external work, and the central proof chain from wires to batching to complexity control is internally consistent. The main text is explicit that several arguments are sketches, with fuller (though in places still terse) versions in the appendix.

minor comments (5)

[Appendix, Lemma 9] The proof of Lemma 9 in the appendix gives a detailed argument only for the case where p and q both lie on vertical segments; the two remaining cases (p on a horizontal and q on a vertical segment, and both on horizontal segments) are only set up and dismissed as analogous. Since the contradiction in the first case relies on vertical segments pr and qs, the analogous arguments for the other cases are not immediate and should be spelled out.
[Appendix, Lemma 2] The shortcut argument in the proof of Lemma 2 is the least formal step of the wire construction: the sentence 'Disconnect all lr-wires at the gates of R... All wires crossing R have been shortened by this' asserts both that the reconnection preserves the equivalence pattern and that every affected wire is strictly shortened. Please expand this step to justify explicitly that reconnecting along L cannot change the crossing order with edges or wires and that the replacement strictly shortens w and all other affected wires.
[Section 5.1, Lemma 10] The statement 'As a corollary of Lemma 4, any slides along links of the wires outside of the ε-band do not affect the k-windmill next to e' is not immediate, because Lemma 4 concerns inversions of vertices, not the geometry of windmills. The connection should be explained.
[Section 3, Lemma 3] The proof of Lemma 3 invokes 'Lemma 2 from [11]' for the fact that intersecting links have the same spirality. Since this fact is load-bearing for the transfer from lr-wires to tb-wires, please state the lemma explicitly or give the precise reference.
[Section 2, Notation] The notation Γi−−▶Γj and Γi =▶Γj is unusual and slightly hard to parse; consider defining it more prominently or using a more standard arrow notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the O(n)-wire construction and batching lemmas are proved from wire combinatorics; the cited morph machinery in [11] is prior published work used as a black box, not a fitted or definitional input.

full rationale

The paper's central new claims—existence of equivalent wire sets with spirality O(n) for disconnected graphs (Lemma 2 and Lemma 3) and batching of iterations into s+1 linear morphs with linear complexity (Sections 4-5)—are argued from the geometry of wires, spirality, and slides rather than assumed as inputs. Lemma 2 proves the O(n) spirality bound by induction and a minimality/shortcut argument on layers; the proof does not presuppose the theorem it is establishing. Lemma 8 and Lemma 9 show that one spirality unit can be removed with a single linear morph by constructing a monotone wire grid and interpolating cell-wise; these are not defined in terms of the number of morphs. The paper does rely heavily on [11] (Theorem 8 and several lemmas) for the underlying morph algorithm, and those references are self-citations by overlapping authors. However, they are prior peer-reviewed, parameter-free statements with fixed assumptions; they are not fitted to the data of this paper and do not include the target disconnected-graph O(n) bound as an assumption. Under the reviewing rules, such citations are real evidence and do not by themselves make the derivation circular. No equation in the paper reduces to another by construction, and no fitted parameter is renamed as a prediction. The few abbreviated arguments in the main text (notably the Lemma 2 layer-shortcut) are terse but are expanded in the appendix and do not amount to assumption of the conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters and no new physical entities; wires and spirality are imported from prior work [11]. The proof relies on standard planar topology, the unified straight-line drawing model, the correctness of the prior morph theorem, and generic-position epsilon-band assumptions. The load-bearing combinatorial assumption is Lemma 3, that an equivalent set of wires with O(n) spirality always exists for disconnected inputs.

assumptions (5)

domain assumption Equivalence of ΓI and ΓO implies an equivalent set of wires exists in ΓI with matching crossing order.
Section 2, used as the starting point for all wire constructions; it follows from the homeomorphism between the two drawings.
domain assumption Bends and edge subdivisions are unified into straight-line orthogonal drawings on n vertices before morphing.
Section 2: every edge becomes a single horizontal or vertical segment in both drawings, and n counts vertices plus bends.
domain assumption Prior results from van Goethem and Verbeek [11], especially the O(s) morph for connected drawings and Lemma 9 and Lemma 12 on wire slides, are correct.
Used in Sections 3 and 4 as black boxes; no independent proof is reproduced in this paper.
domain assumption Generic position: small epsilon-bands around edges contain no vertices or bends, so windmill rerouting can be performed without creating crossings.
Section 5.1 introduces this epsilon-band assumption; it is standard in orthogonal drawing arguments.
standard math Jordan curve theorem and planarity of drawing and wires justify the region and crossing-count contradictions used in the proofs.
Used implicitly in the appendices to Lemmas 2, 5, 8, and 9 when arguing about enclosed regions and crossing patterns.

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Pith. "Pith review of Optimal Morphs of Planar Orthogonal Drawings II." pith.science (2026). https://pith.science/paper/VHQBYQWL

@misc{pith2026190808365,
  author       = {Pith},
  title        = {Pith review of: Optimal Morphs of Planar Orthogonal Drawings II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VHQBYQWL}},
  note         = {Machine review of arXiv:1908.08365}
}

abstract

Van Goethem and Verbeek recently showed how to morph between two planar orthogonal drawings $\Gamma_I$ and $\Gamma_O$ of a connected graph $G$ while preserving planarity, orthogonality, and the complexity of the drawing during the morph. Necessarily drawings $\Gamma_I$ and $\Gamma_O$ must be equivalent, that is, there exists a homeomorphism of the plane that transforms $\Gamma_I$ into $\Gamma_O$. Van Goethem and Verbeek use $O(n)$ linear morphs, where $n$ is the maximum complexity of the input drawings. However, if the graph is disconnected their method requires $O(n^{1.5})$ linear morphs. In this paper we present a refined version of their approach that allows us to also morph between two planar orthogonal drawings of a disconnected graph with $O(n)$ linear morphs while preserving planarity, orthogonality, and linear complexity of the intermediate drawings. Van Goethem and Verbeek measure the structural difference between the two drawings in terms of the so-called spirality $s = O(n)$ of $\Gamma_I$ relative to $\Gamma_O$ and describe a morph from $\Gamma_I$ to $\Gamma_O$ using $O(s)$ linear morphs. We prove that $s+1$ linear morphs are always sufficient to morph between two planar orthogonal drawings, even for disconnected graphs. The resulting morphs are quite natural and visually pleasing.

Figures

Figures reproduced from arXiv: 1908.08365 by the authors.

**Figure 2.** A drawing (black) with vertices (open marks) and bends (closed marks). (a) A zigzag-eliminating slide with center link βγ. (b) Introducing two additional bends in a crossing segment ensures orthogonality. (c) A bend-introducing slide. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** (a) A (straight-line) wire w in ΓO (red) and two possible wires in ΓI from different homotopy classes that both match w. (b) A graph with three connected components. (c) Wires in ΓI that cross three times. (d) Set of wires equivalent to ΓO. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: (a) Lemma 1 for a link `→ and sequence S = (`3, `5, `13, `15, `→). (b) The i-core of a spiral for a link ` i ∈ S (gray). (c) The i-layer of the spiral (gray). (d) A layer cannot only contain wires as then we can shorten all wires. i-core for S (for 4 ≤ i ≤ s and i (mod…

**Figure 5.** Figure 5: Regions surrounding `s in Γi−1 and the matching regions in Γi. SL SR Γa L↓ Γb L→ [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: (a) Two points p and q on vertical segments of the drawing that are inverted along both axes imply wires in ΓI that are not equivalent to ΓO. (b) Points p and q on a horizontal and vertical segment. (c) Points p and q on horizontal segments. Proof. (Sketch) The partiti…

**Figure 8.** Figure 8: (a) An ε-band adjacent to the edge. (b) Inserting an s-windmill. (c-d) Reroute wires after linear slide without introducing new crossings. We reroute the wires in W↓ and W→ as follows. Consider an edge e that is crossed by at least two wires in ΓI . By Lemma 9 from [11…

**Figure 9.** Figure 9: (a) A 6ε-box surrounding a vertex v (dashed) with four redrawn edges. (b) Original drawing, rerouted drawing, and straightening the drawing. Drawing Γ r s−1 compared to Γs contains two additional bends in each edge crossed by maximum absolute spirality links in Γs. We …

**Figure 10.** Figure 10: (a) The first link `m from w[i − 1] crossing L below `i must cross L lower than the lowest link `j from S crossing L. (b) There are exactly two configurations for link `m (m < i) crossing directly below left-oriented link `i (for partial wire w[i − 1]). Either the spi…

**Figure 11.** Figure 11: (a) Motion of the vertices in V (gray) defined by the (horizontally extended) zigzag containing `. (b) To change the order of v and w along the y-axis, both must be in the horizontal strip defined by ` (blue) and separated by `. 17 [PITH_FULL_IMAGE:figures/full_fig_p…

**Figure 12.** Figure 12: (a) Regions surrounding `s in Γi−1 and the matching regions in Γi. Regions A, B, C, D are maintained between the two drawings. (b) A vertical slide may merge two links from staircase S in Γi−1 but the result is also a valid staircase. (c) If S is split in Γi−1 then it…

**Figure 13.** Figure 13: (a) Sets SL and SR in Γa and Γb. (b) A vertical edge e ∈ SR cannot cross L→ left of a vertical edge f ∈ SL as vertex u must be x- and y-inverted with one of the endpoints of f during the morph. (c) The y-monotone line cannot cross the edges in the wrong order as then …

**Figure 14.** Figure 14: (a) The number of crossings of two y-monotone tb-wires that cross at least twice can be reduced as the areas in between them (light blue) cannot contain any vertices. (b) Each enclosed region contains an xy-monotone path π. (c) An x-monotone lr-wire and a y-monotone t…

**Figure 15.** Figure 15: (a) An ε-band adjacent to the edge. (b) Rerouting all crossing wires inside the -band while ensuring the wires always pass the last bend-point of the right-most wire. (c) The wire structure is maintained when the crossing links are reduced. (d-e) After reducing the r…

**Figure 16.** Figure 16: (a) An existing staircase before rerouting (blue).(b) The rerouted wires may extend the staircase but never destroy it. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[11]

Optimal morphs of planar orthogonal drawings

Arthur van Goethem and Kevin Verbeek. Optimal morphs of planar orthogonal drawings. In Proceedings of the 34th International Symposium on Computational Geometry (SoCG 2018) , pages 42:1–42:14, 2018. 13 A Omitted proofs Theorem 2. Let ΓI and ΓO be two uniﬁed planar orthogonal drawings of a (disconnected) graphG, whereΓI has spiralitys. We can morphΓI into ...

work page 2018
[1]

How to morph planar graph drawings.SIAM Journal on Computing , 46(2):824–852, 2017

Soroush Alamdari, Patrizio Angelini, Fidel Barrera-Cruz, Timothy Chan, Gior- dano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Penny Haxell, Anna Lubiw, Maurizio Patrignani, Vincenzo Roselli, Sahil Singla, and Bryan Wilkinson. How to morph planar graph drawings.SIAM Journal on Computing , 46(2):824–852, 2017

work page 2017
[2]

Compatible connectivity augmentation of planar disconnected graphs.Discrete & Computational Geometry, 54(2):459–480, 2015

Greg Aloupis, Luis Barba, Paz Carmi, Vida Dujmović, Fabrizio Frati, and Pat Morin. Compatible connectivity augmentation of planar disconnected graphs.Discrete & Computational Geometry, 54(2):459–480, 2015

work page 2015
[3]

Mor- phing planar graph drawings eﬃciently

Patrizio Angelini, Fabrizio Frati, Maurizio Patrignani, and Vincenzo Roselli. Mor- phing planar graph drawings eﬃciently. InProc. 21st International Symposium on Graph Drawing, pages 49–60, 2013

work page 2013
[4]

Morphing orthog- onal planar graph drawings.ACM Transactions on Algorithms , 9(4):29:1–29:24, 2013

Therese Biedl, Anna Lubiw, Mark Petrick, and Michael Spriggs. Morphing orthog- onal planar graph drawings.ACM Transactions on Algorithms , 9(4):29:1–29:24, 2013

work page 2013
[5]

Morphing planar graphs while preserving edge directions

Therese Biedl, Anna Lubiw, and Michael Spriggs. Morphing planar graphs while preserving edge directions. In Proc. 13th International Symposium on Graph Drawing, pages 13–24, 2006

work page 2006
[6]

Orthogonal graph drawing with inﬂexible edges.Computational Geometry, 55:26–40, 2016

Thomas Bläsius, Sebastian Lehmann, and Ignaz Rutter. Orthogonal graph drawing with inﬂexible edges.Computational Geometry, 55:26–40, 2016

work page 2016
[7]

Deformations of plane rectilinear complexes

Stewart Cairns. Deformations of plane rectilinear complexes. The American Mathematical Monthly, 51(5):247–252, 1944

work page 1944

Show all 11 references

[8]

Spirality and optimal orthogonal drawings

Giuseppe Di Battista, Giuseppe Liotta, and Francesco Vargiu. Spirality and optimal orthogonal drawings. SIAM Journal on Computing , 27(6):1764–1811, 1998

1998
[9]

Upward spirality and upward planarity testing.SIAM Journal on Discrete Mathematics , 23(4):1842–1899, 2009

Walter Didimo, Francesco Giordano, and Giuseppe Liotta. Upward spirality and upward planarity testing.SIAM Journal on Discrete Mathematics , 23(4):1842–1899, 2009

2009
[10]

Deformations of plane graphs

Carsten Thomassen. Deformations of plane graphs. Journal of Combinatorial Theory, Series B , 34(3):244–257, 1983

1983

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

Optimal morphs of planar orthogonal drawings

Arthur van Goethem and Kevin Verbeek. Optimal morphs of planar orthogonal drawings. In Proceedings of the 34th International Symposium on Computational Geometry (SoCG 2018) , pages 42:1–42:14, 2018. 13 A Omitted proofs Theorem 2. Let ΓI and ΓO be two uniﬁed planar orthogonal drawings of a (disconnected) graphG, whereΓI has spiralitys. We can morphΓI into ...

work page 2018

[2] [1]

How to morph planar graph drawings.SIAM Journal on Computing , 46(2):824–852, 2017

Soroush Alamdari, Patrizio Angelini, Fidel Barrera-Cruz, Timothy Chan, Gior- dano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Penny Haxell, Anna Lubiw, Maurizio Patrignani, Vincenzo Roselli, Sahil Singla, and Bryan Wilkinson. How to morph planar graph drawings.SIAM Journal on Computing , 46(2):824–852, 2017

work page 2017

[3] [2]

Compatible connectivity augmentation of planar disconnected graphs.Discrete & Computational Geometry, 54(2):459–480, 2015

Greg Aloupis, Luis Barba, Paz Carmi, Vida Dujmović, Fabrizio Frati, and Pat Morin. Compatible connectivity augmentation of planar disconnected graphs.Discrete & Computational Geometry, 54(2):459–480, 2015

work page 2015

[4] [3]

Mor- phing planar graph drawings eﬃciently

Patrizio Angelini, Fabrizio Frati, Maurizio Patrignani, and Vincenzo Roselli. Mor- phing planar graph drawings eﬃciently. InProc. 21st International Symposium on Graph Drawing, pages 49–60, 2013

work page 2013

[5] [4]

Morphing orthog- onal planar graph drawings.ACM Transactions on Algorithms , 9(4):29:1–29:24, 2013

Therese Biedl, Anna Lubiw, Mark Petrick, and Michael Spriggs. Morphing orthog- onal planar graph drawings.ACM Transactions on Algorithms , 9(4):29:1–29:24, 2013

work page 2013

[6] [5]

Morphing planar graphs while preserving edge directions

Therese Biedl, Anna Lubiw, and Michael Spriggs. Morphing planar graphs while preserving edge directions. In Proc. 13th International Symposium on Graph Drawing, pages 13–24, 2006

work page 2006

[7] [6]

Orthogonal graph drawing with inﬂexible edges.Computational Geometry, 55:26–40, 2016

Thomas Bläsius, Sebastian Lehmann, and Ignaz Rutter. Orthogonal graph drawing with inﬂexible edges.Computational Geometry, 55:26–40, 2016

work page 2016

[8] [7]

Deformations of plane rectilinear complexes

Stewart Cairns. Deformations of plane rectilinear complexes. The American Mathematical Monthly, 51(5):247–252, 1944

work page 1944