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Size in maximal triangle-free graphs and minimal graphs of diameter 2

1995
*
Discrete Mathematics
*

A

doi:10.1016/0012-365x(94)00190-t
fatcat:x7yn3i5frbfr3e5uu5fju4awqm
*diameter**2**graph*is*minimal*if the deletion*of*any edge increases the*diameter*. We show that a*triangle*-*free**graph*is*maximal*if*and*only if it is*minimal**of**diameter**2*. ... A*triangle*-*free**graph*is*maximal*if the addition*of*any edge creates a*triangle*. ... We also say G is an (n, m)*graph**and*has order n*and**size*m. A*maximal**triangle*-*free*(MTF)*graph*G has no*triangles*but the addition*of*any new edge creates a*triangle*. ...##
###
Minimal Prime Graphs of Solvable Groups
[article]

2020
*
arXiv
*
pre-print

*Minimal*prime

*graphs*turn out to be

*of*particular interest,

*and*

*in*this paper we pursue this further by exploring, among other things,

*diameters*, Hamiltonian cycles

*and*the property

*of*being self-complementary ... We also study a new, but closely related notion

*of*

*minimality*for prime

*graphs*

*and*look into counting

*minimal*prime

*graphs*. ... Yong Yang, the director

*of*the REU program, is thanked for persevering despite the conditions created by the pandemic

*and*running the REU program

*in*full. ...

##
###
Hamiltonicity, diameter, domination, packing, and biclique partitions of Mycielski's graphs

1998
*
Discrete Applied Mathematics
*

From [l], a

doi:10.1016/s0166-218x(97)00126-1
fatcat:c5d46mzsljgijcvvfltrhjent4
*graph*is*maximal**triangle*-*free*if*and*only if it is*triangle*-*free**and*has*diameter*at most*2*. ... Since the classes*of**triangle*-*free**graphs*,*maximal**triangle*-*free**graphs*,*and**diameter**2**graphs*are all closed under Mycielski's construction, it is tempting to conjecture that the same is true for*minimal*...##
###
Minimally triangle-saturated graphs: adjoining a single vertex

2002
*
The Australasian Journal of Combinatorics
*

The results are applied to construct primitive

dblp:journals/ajc/EggletonM02
fatcat:cbpfah6tpveubirohnwia3wcoi
*maximal**triangle*-*free**graphs**of*every order n ≥ 9. ... Such a*graph*is*minimally**triangle*-saturated if removal*of*any edge from G leaves a*graph*that is not*triangle*-saturated. ... Acknowledgment: Research for this paper was supported*in*part by University*of*Newcastle Faculty*of*Science*and*Mathematics Research Grant 250-0013. ...##
###
Page 643 of Mathematical Reviews Vol. , Issue 96b
[page]

1996
*
Mathematical Reviews
*

*in*

*maximal*

*triangle*-

*free*

*graphs*

*and*

*minimal*

*graphs*

*of*

*diameter*

*2*. ... The authors prove the following. (1) A

*triangle*-

*free*

*graph*is MTF if

*and*only if it is a

*minimal*

*diameter*-

*2*

*graph*. (

*2*) Let m

*and*n > 5 be positive integers. ...

##
###
Graph-Theoretic Solutions to Computational Geometry Problems
[chapter]

2010
*
Lecture Notes in Computer Science
*

*triangle*-

*free*hyperbolic arrangements) [Ageev, Disc. ... point p: Let q be a point at distance D from p (not necessarily

*in*input), generate the lune from p

*and*q,

*and*find maximum independent set

*in*the bipartite

*graph*

*of*lune. ... structure is often important

*maximal*outerplanar

*graphs*squaregraphs

*and*kinggraphs bipartite

*graphs*intersection

*graphs*

*of*unit circles planar

*graphs*bridgeless 3-regular

*graphs*Much more likely remains ...

##
###
A 4-colour problem for dense triangle-free graphs

2002
*
Discrete Mathematics
*

*Triangle*-

*free*

*graphs*

*of*order n with minimum degree exceeding n=3 satisfy strong structural properties

*in*several respects. ... We conjecture that these

*graphs*are 4-colourable

*and*give an a rmative answer for regular

*maximal*

*triangle*-

*free*

*graphs*satisfying the degree bound. ... Observe that for a

*triangle*-

*free*

*graph*, being

*maximal*

*triangle*-

*free*is equivalent to having

*diameter*at most

*2*. ...

##
###
The dual diameter of triangulations

2018
*
Computational geometry
*

Trivially,

doi:10.1016/j.comgeo.2017.06.008
fatcat:vfckb57yjfag5ij3pi5rnafnai
*in*this case any*maximizing*triangulation has dual*diameter*$n-*2*$. ... We consider triangulations*of*~$\Poly$ that*minimize*or*maximize*the*diameter**of*their dual*graph*. We show that both triangulations can be constructed*in*$O(n^3\log n)$ time using dynamic programming. ... K. was supported*in*part by the ELC project (MEXT KAKENHI No. 24106008). S.L. is Directeur de Recherches du FRS-FNRS. W.M. is supported*in*part by DFG grants MU 3501/1*and*MU 3501/*2*. ...##
###
Approximate closest community search in networks

2015
*
Proceedings of the VLDB Endowment
*

*In*this paper, given an undirected

*graph*G

*and*a set

*of*query nodes Q, we study community search using the k-truss based community model. ... However, most existing studies do not address the "

*free*rider" issue, that is, nodes far away from query nodes

*and*irrelevant to them are included

*in*the detected community. ... Our CTC model

*maximizes*trussness. Furthermore, by

*minimizing*the

*diameter*, it helps remove

*free*riders, thus reducing the

*size*

*in*a principled manner. ...

##
###
To Approximate Treewidth, Use Treelength!

2016
*
SIAM Journal on Discrete Mathematics
*

We prove that

doi:10.1137/15m1034039
fatcat:lmge5fd5anf4bjedhjtt5c6zoe
*in*any*graph*G, the*diameter**of*any*minimal*separator S*in*G is at most (G)/*2*· (|S| − 1), with (G) the length*of*a longest isometric cycle*in*G. ... To this end, we establish new upper-bounds on the*diameter**of**minimal*separators*in**graphs*. ... We wish to thank the referees for their careful reading*of*the first version*of*this manuscript,*and*their useful comments. ...##
###
Approximate Closest Community Search in Networks
[article]

2015
*
arXiv
*
pre-print

*In*this paper, given an undirected

*graph*G

*and*a set

*of*query nodes Q, we study community search using the k-truss based community model. ... However, most existing studies do not address the "

*free*rider" issue, that is, nodes far away from query nodes

*and*irrelevant to them are included

*in*the detected community. ... Our CTC model

*maximizes*trussness. Furthermore, by

*minimizing*the

*diameter*, it helps remove

*free*riders, thus reducing the

*size*

*in*a disciplined manner. ...

##
###
C3 saturated graphs

2005
*
Discrete Mathematics
*

*In*the second part

*of*the paper a useful characterization

*of*

*graphs*which are C 3 -saturated

*and*C 4 -

*free*will be given

*in*terms

*of*the adjacency

*and*incidence matrices A

*and*B. ...

*In*addition to this, a full description

*of*the set

*of*planar C 3 -saturated

*graphs*, PSAT(n, C 3 ), will be given. It will be shown that there are only three kinds

*of*such

*graphs*. ... Furthermore, we denote by ex(n, R)

*and*mx(n, R) the

*maximal*

*and*the

*minimal*

*size*

*of*an R-saturated

*graph*

*of*order n. ...

##
###
Balanced Schnyder woods for planar triangulations: an experimental study with applications to graph drawing and graph separators
[article]

2019
*
arXiv
*
pre-print

*In*this work we consider balanced Schnyder woods for planar

*graphs*, which are Schnyder woods where the number

*of*incoming edges

*of*each color at each vertex is balanced as much as possible. ...

*of*the layout

*of*Schnyder drawings,

*and*provides an efficient tool for computing short

*and*balanced cycle separators. ... Among all Schnyder woods, the

*minimal*(resp.

*maximal*) one plays an important role

*in*many algorithmic

*and*combinatorial problems. ...

##
###
Correlation Clustering with Partial Information
[chapter]

2003
*
Lecture Notes in Computer Science
*

We also prove that this linear program has a gap

doi:10.1007/978-3-540-45198-3_1
fatcat:etjrx42hebbcdpqjbxtdqao4by
*of*Ω(log n),*and*therefore our approximation is tight under this approach. We also give an O(r 3 )-approximation algorithm for Kr,r-minor-*free**graphs*. ... They proved NP-hardness*and*gave constant-factor approximation algorithms for the special case*in*which the*graph*is complete (full information)*and*every edge has weight +1 or −1. ... Many thanks go to Shuchi Chawla, Avrim Blum, Mohammad Mahdian, David Liben-Nowell,*and*Grant Wang. Many results*in*this paper were inspired by conversations with Seffi Naor. ...##
###
Page 3173 of Mathematical Reviews Vol. , Issue 90F
[page]

1990
*
Mathematical Reviews
*

These problems are also solved when the

*graph*is assumed*triangle*-*free**and*when the*graph*is assumed C4-*free*. The result*in*the final case is surprisingly smaller than*in*the other cases. ... Asymptotically sharp bounds are found for both the*maximal**diameter**and*maxi- mal radius*of*such a*graph*. ...
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