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### Each maximal planar graph with exactly two separating triangles is Hamiltonian

Guido Helden
2007 Discrete Applied Mathematics
Chen [Any maximal planar graph with only one separating triangle is Hamiltonian J. Combin.  ...  Optim. 7 (2003) 79-86] proved that any maximal planar graph with only one separating triangle is still Hamiltonian.  ...  Let G be a maximal planar graph with only one separating triangle, then G is Hamiltonian.  ...

### Extension of a theorem of Whitney

Paul C. Kainen, Shannon Overbay
2007 Applied Mathematics Letters
It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle  ...  Since there exist maximal planar graphs which are not Hamiltonian, this is the strongest possible extension. Note that one cannot just add edges without possibly creating triangles that separate.  ...  Every plane graph with no separating triangle is a subgraph of a Hamiltonian plane graph. The approach here, based on Overbay's thesis  , uses connectivity.  ...

### Are highly connected 1-planar graphs Hamiltonian? [article]

Therese Biedl
2019 arXiv   pre-print
We show that this is false in general, even for 5-connected graphs, but true if the graph has a 1-planar drawing where every region is a triangle.  ...  It is well-known that every planar 4-connected graph has a Hamiltonian cycle. In this paper, we study the question whether every 1-planar 4-connected graph has a Hamiltonian cycle.  ...  For any N , there exists a simple maximal 1-plane graph G with n ≥ N vertices that has no Hamiltonian path. Proof.  ...

### Arc diagrams, flip distances, and Hamiltonian triangulations

Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, Manuel Wettstein
2018 Computational geometry
We show that every triangulation (maximal planar graph) on n ≥ 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips.  ...  As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n − 6) edges the resulting  ...  A maximal planar graph on n vertices is is a planar graph with 3n − 6 edges.  ...

### Arc diagrams, flip distances, and Hamiltonian triangulations [article]

Jean Cardinal and Michael Hoffmann and Vincent Kusters and Csaba D. Tóth and Manuel Wettstein
2016 arXiv   pre-print
We show that every triangulation (maximal planar graph) on n> 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips.  ...  As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting  ...  A maximal planar graph on n vertices is a planar graph with 3n − 6 edges.  ...

### Finding Hamiltonian cycles in Delaunay triangulations is NP-complete

Michael B. Dillencourt
1996 Discrete Applied Mathematics
This research was supported in part by the University of California, Irvine through an allocation of computer resources. ' A graph G is k-Hamiltonian if any graph obtained by removing k vertices from G  ...  is Hamiltonian. 0166-218X/96/\$15.00 Q 1996-Elsevier Science B.V.  ...  Acknowledgements It is a pleasure to acknowledge stimulating conversations with Warren Smith and Igor Rivin.  ...

### Alliances and Bisection Width for Planar Graphs [chapter]

Martin Olsen, Morten Revsbæk
2013 Lecture Notes in Computer Science
We show that any planar graph with minimum degree at least 4 can be split into two alliances in polynomial time.  ...  This improves a recently published n + 1 upper bound on the bisection width of planar graphs without separating triangles and supports the folklore conjecture that a general upper bound of n exists for  ...  The expanded graph does not have a separating triangle: In this case the graph is 4-connected since all maximal planar graphs without a separating triangle are 4-connected  and thus contains a hamiltonian  ...

### On Alliance Partitions and Bisection Width for Planar Graphs

Martin Olsen, Morten Revsbaek
2013 Journal of Graph Algorithms and Applications
This improves a recently published n + 1 upper bound on the bisection width of planar graphs without separating triangles and supports the folklore conjecture that a general upper bound of n exists for  ...  We base this on a proof of an upper bound of n on the bisection width for 4-connected planar graphs with an odd number of vertices.  ...  The expanded graph does not have a separating triangle: In this case the graph is 4-connected since all maximal planar graphs without a separating triangle are 4-connected  and thus contains a hamiltonian  ...

### Coupon-Coloring and Total Domination in Hamiltonian Planar Triangulations

Zoltán Lóránt Nagy
2018 Graphs and Combinatorics
We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning  ...  We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings.  ...  Acknowledgement This work started at the Workshop on Graph and Hypergraph Domination in Balatonalmádi in June 2017.  ...

### Computing Cartograms with Optimal Complexity [article]

Md. Jawaherul Alam, Therese Biedl, Stefan Felsner, Michael Kaufmann, Stephen G. Kobourov, Torsten Ueckerdt
2011 arXiv   pre-print
The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs.  ...  Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs.  ...  Finally, we showed that if the graph admits a one-legged Hamiltonian cycle (for example outer-planar graphs), only 6 sides are needed.  ...

### Embedding Vertices at Points: Few Bends Suffice for Planar Graphs [chapter]

Michael Kaufmann, Roland Wiese
2004 Graph Algorithms and Applications 3
Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge.  ...  Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend.  ...  Then, in the following section we present a small plane graph with only 12 vertices without any external hamiltonian cycle and a point set, and we prove that any planar drawing of this graph on this point  ...

### Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

Michael Kaufmann, Roland Wiese
2002 Journal of Graph Algorithms and Applications
Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge.  ...  Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend.  ...  Then, in the following section we present a small plane graph with only 12 vertices without any external hamiltonian cycle and a point set, and we prove that any planar drawing of this graph on this point  ...

### Embedding Vertices at Points: Few Bends Suffice for Planar Graphs [chapter]

Michael Kaufmann, Roland Wiese
1999 Lecture Notes in Computer Science
Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge.  ...  Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend.  ...  Then, in the following section we present a small plane graph with only 12 vertices without any external hamiltonian cycle and a point set, and we prove that any planar drawing of this graph on this point  ...

### Page 48 of Mathematical Reviews Vol. , Issue 80A [page]

1980 Mathematical Reviews
On the connectivity of maximal planar graphs. J. Graph Theory 2 (1978), no. 4, 307-314. The vertex connectivity of a maximal planar graph (every face is a triangle) is known to be at least 3.  ...  A minimal vertex cut set in a maximal planar graph must form a cycle in the graph (a separating cycle).  ...

### Computing Cartograms with Optimal Complexity

Md. Jawaherul Alam, Therese Biedl, Stefan Felsner, Michael Kaufmann, Stephen G. Kobourov, Torsten Ueckerdt
2013 Discrete & Computational Geometry
The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs.  ...  Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs. * This research was initiated at the Dagstuhl  ...  Figure 4 : 4 (a) A maximal planar Hamiltonian graph with a weight function that requires at least one 8-sided polygon in any cartogram.  ...
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