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4-connected maximal planar graphs are 4-ordered
2002
Discrete Mathematics
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It is shown that in a 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. ...
doi:10.1016/s0012-365x(02)00438-7
fatcat:dyint4tczvftbpnfy4zc62zarm
Tight bounds on maximal and maximum matchings
2004
Discrete Mathematics
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In this paper, we study lower bounds on the size of 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. ...
doi:10.1016/j.disc.2004.05.003
fatcat:b7ntmc3vxjbg5mrunx7ekhldgy
Tight Bounds on Maximal and Maximum Matchings
[chapter]
2001
Lecture Notes in Computer Science
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In this paper, we study lower bounds on the size of 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. ...
doi:10.1007/3-540-45678-3_27
fatcat:27hzfab5ifbxjbuut7me37pzni
On topological aspects of orientations
2001
Discrete Mathematics
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Using these orientations, we give a characterization of 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 ...
doi:10.1016/s0012-365x(00)00201-6
fatcat:hg7satxcq5ggnbn7siuns7xjmy
Bounds of Eigenvalues of -Minor Free Graphs
2009
Journal of Inequalities and Applications
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B Let G be a simple 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. ...
doi:10.1155/2009/852406
fatcat:ebikld2ldrantbzu5hahp63oea
(3,3)-Linked planar graphs
2008
Discrete Mathematics
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We shall prove that a 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] . ...
doi:10.1016/j.disc.2007.09.013
fatcat:2tj3ezsitnenfet6jjm7v5wmym
Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph
[article]
2012
arXiv
pre-print
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and proves both the four color conjecture and the uniquely 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. ...
arXiv:0911.1587v3
fatcat:ifjdqgpvjnh7nlitqzr26p2hcq
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
2016
Algorithmica
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In this paper, we present a polynomial-time algorithm to test whether a given 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. ...
doi:10.1007/s00453-016-0200-5
fatcat:i357lisyqzgkdbt7a3cdy3uynm
Constructing Triangle Decomposable Multigraphs with Minimum Multi-edges
[article]
2021
arXiv
pre-print
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We explore several classes of 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 . ...
arXiv:2108.07890v2
fatcat:l75edsvybng4jeakxxsody42ia
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
[chapter]
2014
Lecture Notes in Computer Science
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A 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] . ...
doi:10.1007/978-3-662-45803-7_17
fatcat:55ue3f7zqjb73hs22hkvx6ifg4
On properties of maximal 1-planar graphs
2012
Discussiones Mathematicae Graph Theory
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We study 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. ...
doi:10.7151/dmgt.1639
fatcat:fefldaaa6nhthkowkdpz5f4k7m
Universal Point Subsets for Planar Graphs
[chapter]
2012
Lecture Notes in Computer Science
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and lower bounds for F (n, G) for different families of 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. ...
doi:10.1007/978-3-642-35261-4_45
fatcat:fuaokaa5obatta2sbwcsgx6m2e
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
[article]
2014
arXiv
pre-print
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In this paper, we present a polynomial-time algorithm to test whether a given 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 ...
arXiv:1409.0461v1
fatcat:wrv3dopp6nayphbh6u5la7dgam
On the Density of Maximal 1-Planar Graphs
[chapter]
2013
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-642-36763-2_29
fatcat:qnubpcd7xnafpejcpaurapcnaq
The reconstruction of maximal planar graphs. I. Recognition
1981
Journal of combinatorial theory. Series B (Print)
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Let G be a 2-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. ...
doi:10.1016/0095-8956(81)90063-0
fatcat:vsxxjqbk3neg3j7pqdbjmalypm
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