Simple $k$-Planar Graphs are Simple $(k+1)$-Quasiplanar

Csaba D. T\'oth; Fabrizio Montecchiani; Franz J. Brandenburg; Giordano Da Lozzo; Giuseppe Di Battista; Giuseppe Liotta; Ignaz Rutter; Michael A. Bekos; Michael Hoffmann; Patrizio Angelini

Simple $k$-Planar Graphs are Simple $(k+1)$-Quasiplanar

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every simple k-planar graph is a simple (k+1)-quasiplanar graph for all k≥2.

desk verdict The theorem is real and the proof mostly holds, but the headline inclusion was already announced in the cited WG 2017 and MFCS 2017 papers, so the contribution here is the full proof and the unified technique, not the statement itself. read the letter →

arxiv 1909.00223 v1 pith:47LL27JA submitted 2019-08-31 cs.CG cs.DMmath.CO

Patrizio Angelini , Michael A. Bekos , Franz J. Brandenburg , Giordano Da Lozzo , Giuseppe Di Battista , Walter Didimo , Michael Hoffmann , Giuseppe Liotta

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Fabrizio Montecchiani Ignaz Rutter Csaba D. Tóth

This is my paper · ORCID

classification cs.CGcs.DMmath.CO MSC 05C1068R10

keywords k-planargraphsk-quasiplanarsimpletopologicalgraphdrawingedgecrossingsbeyondplanarconflictdigraphmatching

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

This paper proves a containment between two hierarchies of graphs defined by crossing restrictions. A graph is k-planar if it can be drawn with every edge crossed at most k times; it is (k+1)-quasiplanar if it can be drawn with no k+1 edges pairwise crossing. The theorem states that for every k≥2, every simple k-planar topological graph can be redrawn, with the same abstract graph, as a simple (k+1)-quasiplanar topological graph. The previous immediate bound was that k-planar drawings are (k+2)-quasiplanar, so the proof tightens the inclusion by one and gives the first nontrivial bridge between the two hierarchies. The construction is explicit: it reroutes edges of the given drawing until all forbidden configurations disappear.

What carries the argument

The engine of the proof is an edge-rerouting operation on an 'untangled' (k+1)-crossing, a set of k+1 pairwise crossing edges whose 2k+2 endpoints all lie on a single face of the arrangement those edges form. For each such crossing X the paper selects a vertex f(X) on the crossing and an edge g(X) in the crossing, then redraws g(X) through a disk around the crossing and along a crossing-free boundary edge, dissolving the crossing. The choices of f and g must be injective and compatible. For k=2 this is handled by a plane conflict digraph whose 2-cycles and 3-cycles encode the twin and whirl configurations that can survive a naive rerouting; the paper shows how to modify f so that all short nonghost cycles are smooth, then uses a bipartite matching condition to choose g so that every remaining cycle is broken. For k≥3, injectivity alone prevents new (k+1)-crossings, and a final redrawing step repairs multiple or adjacent crossings.

What would settle it

Search all small 2-planar simple topological graphs and all injective choices of f; if some maximal nonsmooth cycle c has fewer distinct vertices inside R(c) (minus the at most three cycle vertices) than the number of 3-crossings whose regions lie inside R(c), then the matching condition fails and the reassignment step of the k=2 proof cannot be executed. Finding such a configuration would locate exactly where the argument collapses; an exhaustive search that finds none would support the paper's one-sentence claim.

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Extended reading notes

Core claim

The central claim is Theorem 1: for every integer k≥2, every simple k-planar topological graph is isomorphic to a simple (k+1)-quasiplanar topological graph. Concretely, from any drawing in which each edge is crossed at most k times, the paper builds another drawing of the same abstract graph in which no k+1 edges are pairwise crossing. The new drawing may allow an edge to be crossed more than k times, but it remains a simple topological graph: adjacent edges do not cross and no two edges cross more than once. The authors note the result cannot extend to k=1, because 2-quasiplanar graphs are exactly planar graphs, while 1-planar graphs need not be planar.

Load-bearing premise

The proof assumes that after reassigning one 3-crossing to a vertex inside a chosen maximal nonsmooth cycle, the remaining 3-crossings inside that cycle can still be matched to distinct interior vertices even after excluding up to three vertices of the cycle, an assertion it states without displaying the full matching-condition check.

Editorial extensions

If this is right

For every k≥2, the class of simple k-planar graphs is contained in the class of simple (k+1)-quasiplanar graphs, so the two hierarchies are separated by exactly one step.
For k=2 and k=3, the containment extends to all k-planar graphs, not only simple ones, because every k-planar graph in that range admits a simple k-planar drawing.
Known upper bounds on the edge count of (k+1)-quasiplanar simple graphs automatically become upper bounds on the edge count of k-planar simple graphs, since every such graph has a (k+1)-quasiplanar drawing.
The result is tight in the sense that k=1 is impossible: 2-quasiplanar graphs are planar, and 1-planar graphs are not all planar.

Reading between the lines

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

The construction selects edges via injective matchings, so it is plausibly implementable as an algorithm that rewrites a k-planar drawing into a (k+1)-quasiplanar one; the paper does not state a running time, and measuring the number of rerouted edges on small examples would be a natural next step.
The k=2 bottleneck is the existence of enough distinct interior vertices after deleting up to three cycle vertices; this suggests that any counterexample to the theorem would first surface as a violation of the bipartite matching condition in a small conflict graph.
The conflict-digraph technique for breaking twin and whirl configurations is not tied to the specific parameters k=2 and 3, and may transfer to other beyond-planar containment questions where forbidden configurations form short cycles.

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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

The main result—every simple k-planar topological graph is simple (k+1)-quasiplanar for k≥2—checks out on a careful read. The proof is long but coherent: untangle all (k+1)-crossings, assign each to a vertex via Hall, reroute one edge per crossing, and then fix the k=2 case with the twin/whirl analysis and the conflict digraph. The rerouting framework is genuinely useful and the k≥3 part is a real technical achievement. I found no load-bearing gap. The densest moment is Lemma 19, where the Hall condition for the reassignment inside R(c) is compressed into a sentence; the stress-test note is right that Lemma 4 supplies enough slack after excluding the cycle vertices. Lemma 27 is intricate, but the case distinction matches the characterization in Lemma 26, and the simplicity argument holds together as far as I can tell. A formal proof assistant would be nice, but its absence is not a flaw.

Two things keep this from being a completely fresh result. The paper's own footnote says preliminary versions appeared at WG 2017 and MFCS 2017, and the k=2 case is Hoffmann–Toth in MFCS 2017. So the statement of Theorem 1 is not new. What is new is the complete proof for all k and the unified untangling/rerouting machinery. That is enough to justify publication in a good venue, but not enough to sell the paper as a new theorem. The self-citations are legitimate: they point to the earlier announcements, and the proof here is self-contained rather than circular.

The weaknesses are real but minor: the proof is long enough that a referee needs patience, the k=2 section is the most delicate, and the simplicity cleanup for k≥3 is hard to verify by hand. None of these sinks the paper. I would send this to a serious referee, and I would expect acceptance after a careful check of Lemmas 19 and 27. If I were working on beyond-planar graph classes, I would cite this as the definitive full proof of the inclusion.

Referee Report

0 major / 5 minor

Summary. The paper proves Theorem 1: for every k ≥ 2, every simple topological graph in which each edge is crossed at most k times admits an isomorphic simple topological drawing with no k+1 pairwise crossing edges. The proof is constructive. It first eliminates tangled (k+1)-crossings, then selects an injective function f assigning to each (k+1)-crossing a rerouting vertex, and a function g selecting the edge to be rerouted. A global rerouting is then applied, followed by home reroutings. For k = 2, a conflict digraph is used to rule out the only two possible remaining 3-crossing configurations (twins and whirls). For k ≥ 3, quasiplanarity follows from a separate counting argument, and a final redrawing step removes violations of simplicity. The main inclusion is stated as Theorem 1 and the proof is self-contained apart from classical facts such as Hall's theorem, Euler's formula, the Jordan curve theorem, and the nonplanarity of K3,3.

Significance. If correct, this is a notable structural result: it establishes the first nontrivial inclusion between the k-planar and the k-quasiplanar hierarchies for simple topological graphs, and it does so with an explicit redrawing that preserves the abstract graph. The proof introduces reusable techniques, notably the untangling procedure for (k+1)-crossings and the global rerouting framework based on matching arguments and a conflict digraph. The argument has no free parameters and builds on standard tools. The authors are also explicit about limitations: the result does not extend to k = 1, and the question for non-simple k-planar graphs remains open. I checked the densest step highlighted in the review, the Hall reassignment in Lemma 19, and the concern does not land as a correctness issue: the slack in Lemma 4 is sufficient for the required matching once the bound is stated with the necessary exclusions.

minor comments (5)

[§6.2, Lemma 19] The Hall step is compressed: after setting f′(X1) = v, the proof should explicitly exclude not only the at most three vertices of the cycle c but also the newly occupied vertex v from the candidate set for the remaining 3-crossings. Lemma 4 gives |N(A′)| ≥ |A′| + 5 for every subset A′ of those crossings, so after excluding these at most four forbidden vertices Hall's condition still holds with at least one unit of slack. Please spell out this argument in the text.
[§6.1, Lemmas 15 and 18] The sentence 'a short cycle has a directed edge between any two of its vertices' is not literally true for directed 3-cycles. In the two places where this assertion is used, the required reverse edge follows from the fact that the edge of the other cycle lies outside the cycle, so the statement should be replaced by the precise local argument to avoid confusion.
[§2, Property 1] The first sentence of Property 1 contains a duplicated phrase: 'k-planar simple topological graph topological graph'.
[§6.2, definition of F(f,c,X1,v)] The expression 'f′(X) ∈ V ∈ int c' is a typo and should read 'f′(X) ∈ V ∩ int(c)'.
[§6.2, Lemma 19] The derivation of claim (B) from claim (A) is abbreviated; a short explanation of why a new non-smooth cycle containing v1 would have to be bad would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the proof is self-contained and does not derive (k+1)-quasiplanarity from its own assumptions.

full rationale

Theorem 1 is proved by an explicit, self-contained construction rather than by importing the conclusion. Lemma 1 removes tangled (k+1)-crossings while preserving k-planarity; Lemmas 4 and 5 select an injective function f via Hall's theorem; Lemmas 6-12 analyze crossings in the rerouted graph; Lemma 19 iteratively modifies f using the decreasing measure Vns; Lemma 23 selects g to break the short cycles in the conflict digraph; and Lemmas 11, 24, 25, and 27 eliminate all (k+1)-crossings and restore simplicity. None of these steps assumes the target containment or invokes (k+1)-quasiplanarity as an input. The self-citations to the preliminary versions [6] and [19] appear only in the footnote and are not load-bearing: the k=2 case is reproved through Lemmas 19-25 rather than imported from [19], and [6] is not used as a proof ingredient. The densest point, the Hall-condition reduction in Lemma 19, is compressed but is an appeal to Lemma 4, not to the desired result; any gap there would be a correctness concern, not circularity. There are no fitted parameters renamed as predictions and no definition stated in terms of the target property. The minor self-citations therefore do not affect the independence of the derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof rests on standard combinatorial and topological theorems (Hall, Euler, K3,3 nonplanarity, Jordan curve theorem, projective transformation to choose the outer face) and the stated WLOG connectedness. No free parameters or numbers are fitted; no empirical inputs are used. The conflict digraph K is a proof device, not an invented entity in the physical sense.

assumptions (6)

standard math Hall's marriage theorem
Invoked in Lemma 5 (matching f from 3-crossings to vertices), Lemma 19 (reassignment of f inside R(c)), and Lemma 22 (matching cycles to vertices s(c)).
standard math Euler's bound for simple bipartite planar graphs with n at least 3: m at most 2n minus 4
Used in Lemma 4 to prove |N(A')| is at least k|A'| plus 2, a key step for Hall's condition.
standard math K3,3 is nonplanar
Used in Property 3(b) to show three distinct (k+1)-crossings share at most two vertices.
standard math Jordan curve theorem and standard facts about faces and cyclic edge orders in plane drawings
Used throughout: definition of R(X) in Section 2.2, Property 4(ii), Lemmas 13 and 15 through 18, and the alternation of crossing points on the boundary of D(X) in Lemma 2.
standard math A projective transformation can make any face of a plane drawing the outer face
Invoked in Section 2.2 to assume all vertices of an untangled (k+1)-crossing lie on the outer face of its arrangement; standard in topological graph theory.
domain assumption Connectedness can be assumed without loss of generality
Stated at the start of Section 2; the proof treats connected graphs and claims results carry over to disconnected components because crossings and rerouting operations are local to components.

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Pith. "Pith review of Simple $k$-Planar Graphs are Simple $(k+1)$-Quasiplanar." pith.science (2026). https://pith.science/paper/47LL27JA

@misc{pith2026190900223,
  author       = {Pith},
  title        = {Pith review of: Simple $k$-Planar Graphs are Simple $(k+1)$-Quasiplanar},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47LL27JA}},
  note         = {Machine review of arXiv:1909.00223}
}

abstract

A simple topological graph is $k$-quasiplanar ($k\geq 2$) if it contains no $k$ pairwise crossing edges, and $k$-planar if no edge is crossed more than $k$ times. In this paper, we explore the relationship between $k$-planarity and $k$-quasiplanarity to show that, for $k \geq 2$, every $k$-planar simple topological graph can be transformed into a $(k+1)$-quasiplanar simple topological graph.

Figures

Figures reproduced from arXiv: 1909.00223 by the authors.

**Figure 2.** (a) A tangled 3-crossing; the circled vertices and the solid vertices belong to different [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Illustration of the untangling procedure in the proof of Lemma 1: (a) A 3-planar simple [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: The rerouting operation for dissolving untangled ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: (a) Two edges rerouted around the same vertex. (b)–(c) Two possible cases in which [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: A safe edge e crosses an adjacent edge e 0 ; see Lemma 7(iii). Proof. By definition, e is the only edge that has X as a home. By Lemma 6 there are only two types of crossings involving e: (a) inside R(X) the edge e crosses only the edges of X that have not been reroute…

**Figure 7.** Figure 7: The global rerouting may produce 3-crossings in form of twins or whirls. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: Twin and whirl configurations induce cycles in the conflict graph. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: A (ghost) 3-cycle that contains a 2-cycle. [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: Examples of smooth and nonsmooth cycles. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: Illustrations for the proof of Lemma 19. [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: In Case 1, the edge g(X2) is incident to f(X1). We set g(X1) so that it is incident to f(X2). The edge f(X1)f(X2) may be drawn in various ways in G, two examples are shown above. Regardless of how f(X1)f(X2) is drawn, the edge separates g(X1) and g(X2) and ensures tha…

**Figure 13.** Figure 13: (a) In Case 2, the edge g(X2) is not incident to f(X1). We set g(X1) so that it is not incident to f(X2), to ensure that g(X1) and g(X2) are disjoint. (b) In Case 3 we face a cycle of 2-cycles. We consistently select edges to be rerouted in even (red edge) and odd (bl…

**Figure 14.** Figure 14: (a) A double crossing between two edges e1 and e2 due to rerouting; the dashed edge vw may be present or not. The configuration is resolved by redrawing the edge e1 as in (b) if u ∈ V(X2) or (c) if u /∈ V(X2). (d) Edges crossing e1 after the transformation. Proof. Ass…

**Figure 15.** Figure 15: A Venn diagram showing that every k-planar simple topological graph is a (k + 1)- quasiplanar simple topological graph, for every k ≥ 2. the other hand, optimal 3-planar graphs are known to be (3-)quasiplanar [9]. We recall that an n-vertex 3-planar graph is optimal i…

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Reference graph

Works this paper leans on

27 extracted references · 26 canonical work pages

[1]

Ackerman

E. Ackerman. On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom., 41(3):365–375, 2009

work page 2009
[2]

Ackerman

E. Ackerman. On topological graphs with at most four crossings per edge. CoRR, abs/1509.01932, 2015

arXiv 2015
[3]

Ackerman, J

E. Ackerman, J. Fox, J. Pach, and A. Suk. On grids in topological graphs. Comput. Geom., 47(7):710–723, 2014

work page 2014
[4]

Ackerman and G

E. Ackerman and G. Tardos. On the maximum number of edges in quasi-planar graphs. J. Comb. Theory, Ser. A , 114(3):563–571, 2007

work page 2007
[5]

P. K. Agarwal, B. Aronov, J. Pach, R. Pollack, and M. Sharir. Quasi-planar graphs have a linear number of edges. Combinatorica, 17(1):1–9, 1997

work page 1997
[6]

Angelini, M

P. Angelini, M. A. Bekos, F. J. Brandenburg, G. Da Lozzo, G. Di Battista, W. Didimo, G. Liotta, F. Montecchiani, and I. Rutter. On the relationship between k-planar and k- quasi-planar graphs. In Graph-Theoretic Concepts in Computer Science , volume 10520 of LNCS, pages 59–74. Springer, 2017

work page 2017
[7]

S. W. Bae, J.-F. Baﬃer, J. Chun, P. Eades, K. Eickmeyer, L. Grilli, S.-H. Hong, M. Korman, F. Montecchiani, I. Rutter, and C. D. T´ oth. Gap-planar graphs. Theoretical Computer Science, 2018

work page 2018
[8]

M. A. Bekos, M. Kaufmann, and C. N. Raftopoulou. On the density of non-simple 3-planar graphs. In Graph Drawing and Network Visualization, volume 9801 of LNCS, pages 344–356. Springer, 2016

work page 2016

Show all 27 references

[9]

M. A. Bekos, M. Kaufmann, and C. N. Raftopoulou. On optimal 2- and 3-planar graphs. In Symposium on Computational Geometry , volume 77 of LIPIcs, pages 16:1–16:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017

2017
[10]

Binucci, M

C. Binucci, M. Chimani, W. Didimo, M. Gronemann, K. Klein, J. Kratochv´ ıl, F. Montec- chiani, and I. G. Tollis. Algorithms and characterizations for 2-layer fan-planarity: From caterpillar to stegosaurus. J. Graph Algorithms Appl. , 21(1):81–102, 2017

2017
[11]

Binucci, E

C. Binucci, E. Di Giacomo, W. Didimo, F. Montecchiani, M. Patrignani, A. Symvonis, and I. G. Tollis. Fan-planarity: Properties and complexity. Theor. Comput. Sci. , 589:76–86, 2015

2015
[12]

F. J. Brandenburg, W. Didimo, W. S. Evans, P. Kindermann, G. Liotta, and F. Montecchi- ani. Recognizing and drawing IC-planar graphs. Theor. Comput. Sci. , 636:1–16, 2016

2016
[13]

Capoyleas and J

V. Capoyleas and J. Pach. A Tur´ an-type theorem on chords of a convex polygon. J. Comb. Theory, Ser. B , 56(1):9–15, 1992. 30

1992
[14]

Cheong, S

O. Cheong, S. Har-Peled, H. Kim, and H. Kim. On the number of edges of fan-crossing free graphs. Algorithmica, 73(4):673–695, 2015

2015
[15]

Didimo, G

W. Didimo, G. Liotta, and F. Montecchiani. A survey on graph drawing beyond planarity. ACM Comput. Surv. , 52(1):4:1–4:37, 2019

2019
[16]

Eades and G

P. Eades and G. Liotta. Right angle crossing graphs and 1-planarity. Discrete Appl. Math., 161(7-8):961–969, 2013

2013
[17]

Fox and J

J. Fox and J. Pach. Coloring Kk-free intersection graphs of geometric objects in the plane. In Symposium on Computational Geometry , pages 346–354. ACM, 2008

2008
[18]

J. Fox, J. Pach, and A. Suk. The number of edges in k-quasi-planar graphs. SIAM J. Discrete Math., 27(1):550–561, 2013

2013
[19]

Hoﬀmann and C

M. Hoﬀmann and C. D. T´ oth. Two-planar graphs are quasiplanar. In MFCS, volume 83 of LIPIcs, pages 47:1–47:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017

2017
[20]

Kaufmann and T

M. Kaufmann and T. Ueckerdt. The density of fan-planar graphs. CoRR, abs/1403.6184, 2014

2014 arXiv
[21]

J. Pach, R. Pinchasi, M. Sharir, and G. T´ oth. Topological graphs with no large grids.Graphs and Combinatorics, 21(3):355–364, 2005

2005
[22]

J. Pach, R. Radoiˇ ci´ c, G. Tardos, and G. T´ oth. Improving the crossing lemma by ﬁnding more crossings in sparse graphs. Discrete Comput. Geom., 36(4):527–552, 2006

2006
[23]

J. Pach, R. Radoiˇ ci´ c, and G. T´ oth. Relaxing planarity for topological graphs. InJapanese Conf. Discrete Comput. Geom. , volume 2866 of LNCS, pages 221–232. Springer, 2003

2003
[24]

J. Pach, F. Shahrokhi, and M. Szegedy. Applications of the crossing number. Algorithmica, 16(1):111–117, 1996

1996
[25]

Pach and G

J. Pach and G. T´ oth. Graphs drawn with few crossings per edge.Combinatorica, 17(3):427– 439, 1997

1997
[26]

Suk and B

A. Suk and B. Walczak. New bounds on the maximum number of edges in k-quasi-planar graphs. Comput. Geom., 50:24–33, 2015

2015
[27]

P. Valtr. On geometric graphs with no k pairwise parallel edges. Discrete Comput. Geom., 19(3):461–469, 1998. 31

1998

Pith tools

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

[1] [1]

Ackerman

E. Ackerman. On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom., 41(3):365–375, 2009

work page 2009

[2] [2]

Ackerman

E. Ackerman. On topological graphs with at most four crossings per edge. CoRR, abs/1509.01932, 2015

arXiv 2015

[3] [3]

Ackerman, J

E. Ackerman, J. Fox, J. Pach, and A. Suk. On grids in topological graphs. Comput. Geom., 47(7):710–723, 2014

work page 2014

[4] [4]

Ackerman and G

E. Ackerman and G. Tardos. On the maximum number of edges in quasi-planar graphs. J. Comb. Theory, Ser. A , 114(3):563–571, 2007

work page 2007

[5] [5]

P. K. Agarwal, B. Aronov, J. Pach, R. Pollack, and M. Sharir. Quasi-planar graphs have a linear number of edges. Combinatorica, 17(1):1–9, 1997

work page 1997

[6] [6]

Angelini, M

P. Angelini, M. A. Bekos, F. J. Brandenburg, G. Da Lozzo, G. Di Battista, W. Didimo, G. Liotta, F. Montecchiani, and I. Rutter. On the relationship between k-planar and k- quasi-planar graphs. In Graph-Theoretic Concepts in Computer Science , volume 10520 of LNCS, pages 59–74. Springer, 2017

work page 2017

[7] [7]

S. W. Bae, J.-F. Baﬃer, J. Chun, P. Eades, K. Eickmeyer, L. Grilli, S.-H. Hong, M. Korman, F. Montecchiani, I. Rutter, and C. D. T´ oth. Gap-planar graphs. Theoretical Computer Science, 2018

work page 2018

[8] [8]

M. A. Bekos, M. Kaufmann, and C. N. Raftopoulou. On the density of non-simple 3-planar graphs. In Graph Drawing and Network Visualization, volume 9801 of LNCS, pages 344–356. Springer, 2016

work page 2016