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Each maximal planar graph with exactly two separating triangles is Hamiltonian

2007
*
Discrete Applied Mathematics
*

Chen [

doi:10.1016/j.dam.2007.03.018
fatcat:npbltdz3lrao5aewhboeeu7l4i
*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

2007
*
Applied Mathematics Letters
*

It

doi:10.1016/j.aml.2006.08.019
fatcat:gktwisje7jdhpaodjnxc2kwyfu
*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 [17] , uses connectivity. ...##
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Are highly connected 1-planar graphs Hamiltonian?
[article]

2019
*
arXiv
*
pre-print

We show that this

arXiv:1911.02153v1
fatcat:v2w2mtwuvbe5xhamignpmimwkm
*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

2018
*
Computational geometry
*

We show that every triangulation (

doi:10.1016/j.comgeo.2017.06.001
fatcat:46wm7d5ttjcrbaq74ifjh3giey
*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]

2016
*
arXiv
*
pre-print

We show that every triangulation (

arXiv:1611.02541v2
fatcat:v2i76nkg4bejziekdmwh2bn6ru
*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

1996
*
Discrete Applied Mathematics
*

This research was supported in part by the University of California, Irvine through an allocation of computer resources. ' A

doi:10.1016/0166-218x(94)00125-w
fatcat:fxnvt4ap6fgrhfzg3zqukj3f6a
*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]

2013
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/978-3-642-36065-7_20
fatcat:5avhxxr3ovfkboitcepipmv6hq
*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 [5] and thus contains a*hamiltonian*...##
###
On Alliance Partitions and Bisection Width for Planar Graphs

2013
*
Journal of Graph Algorithms and Applications
*

This improves a recently published n + 1 upper bound

doi:10.7155/jgaa.00307
fatcat:em3xbrbsdvexnkfhtolxi2zxwa
*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 [5] and thus contains a*hamiltonian*...##
###
Coupon-Coloring and Total Domination in Hamiltonian Planar Triangulations

2018
*
Graphs and Combinatorics
*

We determine this parameter in every

doi:10.1007/s00373-018-1945-1
fatcat:vzey7aakozhkbncvpznzydpjyi
*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. ...##
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Computing Cartograms with Optimal Complexity
[article]

2011
*
arXiv
*
pre-print

The complexity of the cartograms can be reduced to 6 if the

arXiv:1201.0066v1
fatcat:yybofmxb7vbxvdzcwvdnarie64
*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]

2004
*
Graph Algorithms and Applications 3
*

Our results show two algorithms for mapping four-connected plane

doi:10.1142/9789812796608_0005
fatcat:5mmbyksh7jerzdww3dkghbpdbi
*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

2002
*
Journal of Graph Algorithms and Applications
*

Our results show two algorithms for mapping four-connected plane

doi:10.7155/jgaa.00046
fatcat:na25cjehenggzoi5exiboiom2a
*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]

1999
*
Lecture Notes in Computer Science
*

Our results show two algorithms for mapping four-connected plane

doi:10.1007/3-540-46648-7_17
fatcat:ockdz3q6djbuxijgfrxwhb26b4
*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

2013
*
Discrete & Computational Geometry
*

The complexity of the cartograms can be reduced to 6 if the

doi:10.1007/s00454-013-9521-1
fatcat:salphn6onfaltkoq2qe3dj3o5m
*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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