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### 4-connected maximal planar graphs are 4-ordered

Wayne Goddard
2002 Discrete Mathematics
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.  ...

### Tight bounds on maximal and maximum matchings

Therese Biedl, Erik D Demaine, Christian A Duncan, Rudolf Fleischer, Stephen G Kobourov
2004 Discrete Mathematics
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.  ...   , 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]

Therese Biedl, Erik D. Demaine, Christian A. Duncan, Rudolf Fleischer, Stephen G. Kobourov
2001 Lecture Notes in Computer Science
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.  ...   , which studied matchings in triangulated planar graphs such that no matched vertex is incident to a separating triplet.  ...

### On topological aspects of orientations

H. de Fraysseix, P. Ossona de Mendez
2001 Discrete Mathematics
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  ...

### Bounds of Eigenvalues of -Minor Free Graphs

Kun-Fu Fang
2009 Journal of Inequalities and Applications
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.  ...

Ryuichi Mori
2008 Discrete Mathematics
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  .  ...

### Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph [article]

Jin Xu
2012 arXiv   pre-print
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.  ...

### On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

Michael A. Bekos, Sabine Cornelsen, Luca Grilli, Seok-Hee Hong, Michael Kaufmann
2016 Algorithmica
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.  ...

### Constructing Triangle Decomposable Multigraphs with Minimum Multi-edges [article]

C. M. Mynhardt, A. K. Wright
2021 arXiv   pre-print
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 .  ...

### On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs [chapter]

Michael A. Bekos, Sabine Cornelsen, Luca Grilli, Seok-Hee Hong, Michael Kaufmann
2014 Lecture Notes in Computer Science
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  , while maximally dense RAC graphs (i.e., RAC graphs with n vertices and exactly 4n − 10 edges) are 1-planar  . Testing whether a graph is RAC is NP-hard  .  ...

### On properties of maximal 1-planar graphs

Dávid Hudák, Tomas Madaras, Yusuke Suzuki
2012 Discussiones Mathematicae Graph Theory
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.  ...

### Universal Point Subsets for Planar Graphs [chapter]

Patrizio Angelini, Carla Binucci, William Evans, Ferran Hurtado, Giuseppe Liotta, Tamara Mchedlidze, Henk Meijer, Yoshio Okamoto
2012 Lecture Notes in Computer Science
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.  ...

### On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs [article]

Michael A. Bekos and Sabine Cornelsen and Luca Grilli and Seok-Hee Hong and Michael Kaufmann
2014 arXiv   pre-print
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  ...

### On the Density of Maximal 1-Planar Graphs [chapter]

Franz J. Brandenburg, David Eppstein, Andreas Gleißner, Michael T. Goodrich, Kathrin Hanauer, Josef Reislhuber
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.  ...

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

S Fiorini, J Lauri
1981 Journal of combinatorial theory. Series B (Print)
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.  ...
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