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

Curtiss Barefoot, Karen Casey, David Fisher, Kathryn Fraughnaugh, Frank Harary
1995 Discrete Mathematics  
A 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.  ... 
doi:10.1016/0012-365x(94)00190-t fatcat:x7yn3i5frbfr3e5uu5fju4awqm

Minimal Prime Graphs of Solvable Groups [article]

Chris Florez, Jonathan Higgins, Kyle Huang, Thomas Michael Keller, Dawei Shen
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.  ... 
arXiv:2011.10403v1 fatcat:b32grgljnjfl3p3afpxkqev6u4

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

David C. Fisher, Patricia A. McKenna, Elizabeth D. Boyer
1998 Discrete Applied Mathematics  
From [l], a 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  ... 
doi:10.1016/s0166-218x(97)00126-1 fatcat:c5d46mzsljgijcvvfltrhjent4

Minimally triangle-saturated graphs: adjoining a single vertex

Roger B. Eggleton, James Arthur MacDougall
2002 The Australasian Journal of Combinatorics  
The results are applied to construct primitive 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.  ... 
dblp:journals/ajc/EggletonM02 fatcat:cbpfah6tpveubirohnwia3wcoi

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]

David Eppstein
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  ... 
doi:10.1007/978-3-642-11409-0_1 fatcat:rqrljmkgqrb3ddw5nszva6ighi

A 4-colour problem for dense triangle-free graphs

Stephan Brandt
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.  ... 
doi:10.1016/s0012-365x(01)00340-5 fatcat:xs54pjuo6vhhtjehgrxdp2hude

The dual diameter of triangulations

Matias Korman, Stefan Langerman, Wolfgang Mulzer, Alexander Pilz, Maria Saumell, Birgit Vogtenhuber
2018 Computational geometry  
Trivially, 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.  ... 
doi:10.1016/j.comgeo.2017.06.008 fatcat:vfckb57yjfag5ij3pi5rnafnai

Approximate closest community search in networks

Xin Huang, Laks V. S. Lakshmanan, Jeffrey Xu Yu, Hong Cheng
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.  ... 
doi:10.14778/2856318.2856323 fatcat:or7ucrcf6vfqnitdmtgrd5eofu

To Approximate Treewidth, Use Treelength!

David Coudert, Guillaume Ducoffe, Nicolas Nisse
2016 SIAM Journal on Discrete Mathematics  
We prove that 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.  ... 
doi:10.1137/15m1034039 fatcat:lmge5fd5anf4bjedhjtt5c6zoe

Approximate Closest Community Search in Networks [article]

Xin Huang, Laks V.S. Lakshmanan, Jeffrey Xu Yu, Hong Cheng
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.  ... 
arXiv:1505.05956v2 fatcat:73ueznc7kzbmbj4zjdmnxbabs4

C3 saturated graphs

Yehuda Ashkenazi
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.  ... 
doi:10.1016/j.disc.2005.02.016 fatcat:fgcqcyhkvbfifixqij75tlxejq

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

Luca Castelli Aleardi
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.  ... 
arXiv:1908.06688v1 fatcat:fvbxquyxnza5vlub7xercdvpwq

Correlation Clustering with Partial Information [chapter]

Erik D. Demaine, Nicole Immorlica
2003 Lecture Notes in Computer Science  
We also prove that this linear program has a gap 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.  ... 
doi:10.1007/978-3-540-45198-3_1 fatcat:etjrx42hebbcdpqjbxtdqao4by

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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