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4-connected maximal planar graphs are 4-ordered

2002
*
Discrete Mathematics
*

It is shown that in a

doi:10.1016/s0012-365x(02)00438-7
fatcat:dyint4tczvftbpnfy4zc62zarm
*4*-*connected**maximal**planar**graph*there is for any four vertices a, b, c and d, a cycle in the*graph*that contains the four vertices and visits them in the*order*a, b, c and d. ... Let G be a*planar**graph*. (1) If G is 3-*connected*then G is 3-*ordered*. (2) If G is*4*-*ordered*, then G is*maximal**planar*, 3-*connected*, and any cut-set of cardinality 3 isolates a vertex. (3) G is not 5-*ordered*... A*4*-*connected**maximal**planar**graph*is*4*-*ordered*. We prove this result in the next section. It uses a lengthy but standard inductive argument. There is also a*connection*with 2-linked*graphs*. ...##
###
Tight bounds on maximal and maximum matchings

2004
*
Discrete Mathematics
*

In this paper, we study lower bounds on the size of

doi:10.1016/j.disc.2004.05.003
fatcat:b7ntmc3vxjbg5mrunx7ekhldgy
*maximal*and maximum matchings in 3-*connected**planar**graphs*and*graphs*with bounded maximum degree. ... For each class, we give a lower bound on the size of matchings, and show that the bound is tight for some*graph*within the class. ... [*4*] , which studied matchings in triangulated*planar**graphs*such that no matched vertex is incident to a separating triplet. ...##
###
Tight Bounds on Maximal and Maximum Matchings
[chapter]

2001
*
Lecture Notes in Computer Science
*

In this paper, we study lower bounds on the size of

doi:10.1007/3-540-45678-3_27
fatcat:27hzfab5ifbxjbuut7me37pzni
*maximal*and maximum matchings in 3-*connected**planar**graphs*and*graphs*with bounded maximum degree. ... For each class, we give a lower bound on the size of matchings, and show that the bound is tight for some*graph*within the class. ... [*4*] , which studied matchings in triangulated*planar**graphs*such that no matched vertex is incident to a separating triplet. ...##
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On topological aspects of orientations

2001
*
Discrete Mathematics
*

Using these orientations, we give a characterization of

doi:10.1016/s0012-365x(00)00201-6
fatcat:hg7satxcq5ggnbn7siuns7xjmy
*4*-*connected**maximal**planar**graphs*and 3-*connected**planar**graphs*, which leads to simple linear time algorithms (de Fraysseix, Ossona de Mendez, Dagasthul ... We*are*concerned with two classes of*planar**graphs*:*maximal**planar**graphs*(i.e. polyhedral*graphs*, triangulations) and*maximal*bipartite*planar**graphs*(i.e. bipartite*planar**graphs*with quadrilateral faces ... vertices, the articulation pair of H .*4*-*connectivity*of*maximal**planar**graphs*The aim of this section is to relate*4*-*connectivity*of*maximal**planar**graphs*and a cocircuit property of each 3-orientation ...##
###
Bounds of Eigenvalues of -Minor Free Graphs

2009
*
Journal of Inequalities and Applications
*

B Let G be a simple

doi:10.1155/2009/852406
fatcat:ebikld2ldrantbzu5hahp63oea
*connected**graph*with*order*n ≥ 3. If G has no K 3,3 -minor, then λ G ≥ − √ 2n −*4*, where equality holds if and only if G is isomorphic to K 2,n−2 . ... In this paper, we have described the K 3,3 -minor free*graphs*and showed that A let G be a simple*graph*with*order*n ≥ 7. If G has no K 3,3 -minor, then ρ G ≤ 1 √ 3n − 8. ... If G 1 and G 2*are*two*maximal**planar**graphs*with*order*n 1 ≥ 3 and n 2 ≥ 3, respectively, then G 1 ⊕ 2 G 2 is not a*maximal**planar**graph*. Proof. We denote a*planar*embedding of G i by G i still. ...##
###
(3,3)-Linked planar graphs

2008
*
Discrete Mathematics
*

We shall prove that a

doi:10.1016/j.disc.2007.09.013
fatcat:2tj3ezsitnenfet6jjm7v5wmym
*planar**graph*with at least six vertices is (3, 3)-linked if and only if G is*4*-*connected*and*maximal*. ... A*graph*G is (m, n)-linked if for any two disjoint subsets R, B ⊂ V (G) with |R| m and |B| n, G has two disjoint*connected*subgraphs containing R and B, respectively. ... It is easy to see that if a*graph*is*4*-*ordered*, then it is 2-linked, and hence (2, 2)-linked. Goddard proved that every*4*-*connected**maximal**planar**graph*is*4*-*ordered*[2] . ...##
###
Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph
[article]

2012
*
arXiv
*
pre-print

and proves both the four color conjecture and the uniquely

arXiv:0911.1587v3
fatcat:ifjdqgpvjnh7nlitqzr26p2hcq
*4*-colorable*planar**graph*conjecture by mathematical method. ... The famous four color theorem states that for all*planar**graphs*, every vertex can be assigned one of*4*colors such that no two adjacent vertices receive the same color. ... They pointed out that there was one obvious leak on the proof of Ïf a*maximal**planar**graph*has a minimum degree of 5, it is not uniquely*4*-colorable. Ḧere I also would like to thank them deeply. ...##
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On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

2016
*
Algorithmica
*

In this paper, we present a polynomial-time algorithm to test whether a given

doi:10.1007/s00453-016-0200-5
fatcat:i357lisyqzgkdbt7a3cdy3uynm
*graph*is*maximal*outer-fan-*planar*. ... On the negative side, we show that testing fan-*planarity*of a*graph*is NP-hard, for the case where the rotation system (i.e., the cyclic*order*of the edges around each vertex) is given. ... . , v n be the*order*of the vertices around the circle in an outerfan-*planar*drawing of a 3-*connected**graph*G. ...##
###
Constructing Triangle Decomposable Multigraphs with Minimum Multi-edges
[article]

2021
*
arXiv
*
pre-print

We explore several classes of

arXiv:2108.07890v2
fatcat:l75edsvybng4jeakxxsody42ia
*planar**graphs*as well as a class of toroidal*graphs*. ... We study triangle decompositions of*graphs*. ... Suppose n =*4*. Notice that the only*maximal**planar**graph*with*order**4*is K*4*. ...##
###
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
[chapter]

2014
*
Lecture Notes in Computer Science
*

A

doi:10.1007/978-3-662-45803-7_17
fatcat:55ue3f7zqjb73hs22hkvx6ifg4
*graph*is k-*planar*, if it can be embedded in the plane with at most k crossings per edge. Obviously, 1-*planar**graphs**are*also fan-*planar*. ... In this paper, we present a polynomial-time algorithm to test whether a given*graph*is*maximal*outer-fan-*planar*. ... RAC*graphs**are*quasi*planar*[9] , while*maximally*dense RAC*graphs*(i.e., RAC*graphs*with n vertices and exactly 4n − 10 edges)*are*1-*planar*[11] . Testing whether a*graph*is RAC is NP-hard [3] . ...##
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On properties of maximal 1-planar graphs

2012
*
Discussiones Mathematicae Graph Theory
*

We study

doi:10.7151/dmgt.1639
fatcat:fefldaaa6nhthkowkdpz5f4k7m
*maximal*1-*planar**graphs*from the point of view of properties of their diagrams, local structure and hamiltonicity. ... A*graph*is called 1-*planar*if there exists a drawing in the plane so that each edge contains at most one crossing. ... Figure 3 . 3 A PN-*graph*. Figure*4*.*4*Forming a*maximal*1-*planar**graph*from PN-*graph*. Figure 5 . 5 The*maximal*1-*planar*drawing with underlying 1-*planar*non-*maximal**graph*. ...##
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Universal Point Subsets for Planar Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

and lower bounds for F (n, G) for different families of

doi:10.1007/978-3-642-35261-4_45
fatcat:fuaokaa5obatta2sbwcsgx6m2e
*planar**graphs*, including*4*-*connected**planar**graphs*and nested-triangles*graphs*. ... A set S of k points in the plane is a universal point subset for a class G of*planar**graphs*if every*graph*belonging to G admits a*planar*straight-line drawing such that k of its vertices*are*represented ... Note that considering*maximal**planar**graphs*is not a limitation, since any*planar**graph*is a subgraph of a*maximal**planar**graph*. Let G be a*maximal**planar**graph*with n vertices. ...##
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On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
[article]

2014
*
arXiv
*
pre-print

In this paper, we present a polynomial-time algorithm to test whether a given

arXiv:1409.0461v1
fatcat:wrv3dopp6nayphbh6u5la7dgam
*graph*is*maximal*outer-fan-*planar*. ... On the negative side, we show that testing fan-*planarity*of a*graph*is NP-hard, for the case where the rotation system (i.e., the cyclic*order*of the edges around each vertex) is given. ... Algorithm 1 : 3 - 13*Connected**Maximal*Outer-Fan-*Planarity*Input : 3-*connected**graph*G = (V, E), subset E ⊆ E Output : true if and only if G is*maximal*outer-fan-*planar*and has an outer-fan-*planar*drawing ...##
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On the Density of Maximal 1-Planar Graphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

*Maximal*1-

*planar*

*graphs*have at most 4n − 8 edges. We show that there

*are*sparse

*maximal*1-

*planar*

*graphs*with only 45 17 n + O(1) edges. ... With a fixed rotation system there

*are*

*maximal*1-

*planar*

*graphs*with only 7 3 n+O(1) edges. This is sparser than

*maximal*

*planar*

*graphs*. ... The trivial

*maximal*1-

*planar*

*graphs*

*are*the complete

*graphs*for n ≤

*4*. The K

*4*can be drawn with triangular faces or with faces with two and four vertices. Proof. ...

##
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The reconstruction of maximal planar graphs. I. Recognition

1981
*
Journal of combinatorial theory. Series B (Print)
*

Let G be a 2-

doi:10.1016/0095-8956(81)90063-0
fatcat:vsxxjqbk3neg3j7pqdbjmalypm
*connected**planar**graph*which is embedded in the plane E and let r be a circuit of G. ... The*planar**graph*G is*maximal*if all the faces of G*are*3-faces, and outerplanar if it has a face containing all the vertices of G on its boundary. ... FIGURE 2 2 FIGURE 2 THEOREM*4*.*4*Let G be a 3-*connected**graph*with minimum va1enc.v at least*4*. Then G is*maximal**planar*if and only if each G,, has a p(v)representation. ...
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