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Topological containment of the 5-clique minus an edge in 4-connected graphs
[article]

2017
*
arXiv
*
pre-print

The

arXiv:1705.01224v2
fatcat:ta74d3t22bajxlwdzxl6mdipla
*topological*containment problem is known to be polynomial-time solvable for any fixed pattern*graph*H, but good characterisations have been found for only a handful of non-trivial pattern*graphs*. ... One step towards this may be to find a good characterisation of*graphs*that do not*topologically*contain the simpler pattern*graph*K_5^-, obtained by removing a single edge from K_5. ...*In*fact, a family of related results can be observed, for*graphs*of increasing connectivity: • 2-connectivity implies*topological*containment of K 3 ; • 3-connectivity implies*topological*containment of ...##
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Extremal connectivity for topological cliques in bipartite graphs

2006
*
Journal of combinatorial theory. Series B (Print)
*

Given an ε-regular bipartite

doi:10.1016/j.jctb.2005.06.005
fatcat:uwjpfecwlvemdb4c3rtum7yq5u
*graph*(A, B) of density at least d and a set X ⊆ A with |X| ε|A|, there are less than ε|B| vertices*in*B which have at most (d − ε)|X| neighbours*in*X. ... Our main result shows that for bipartite*graphs*this gives the correct asymptotics. ... So if we put all the branch vertices into X, there may not be enough room for all the subdivided edges of our*topological**clique*. ...##
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Large Topological Cliques in Graphs Without a 4-Cycle

2004
*
Combinatorics, probability & computing
*

Mader asked whether every C 4 -free

doi:10.1017/s0963548303005650
fatcat:akwy2bpll5drbdkadaj7obshki
*graph*G contains a subdivision of a complete*graph*whose order is at least linear*in*the average degree of G. ... More generally, we prove that every K s,t -free*graph*of average degree r contains a subdivision of a complete*graph*of order r 1 2 + 1 2(s−1) −o(1) . ... Further results about (*topological*) minors*in**graphs*of large girth can be found*in*[17, 12, 13] . The proof of Theorem 2 uses results of Komlós and Szemerédi [9, 10] . ...##
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A characterization of the geometry of large maximal cliques of the alternating forms graph

2008
*
Innovations in Incidence Geometry Algebraic Topological and Combinatorial
*

We prove that the geometry of vertices, edges and q n -

doi:10.2140/iig.2008.8.81
fatcat:zbgienbtbnf4janpltb23xwwpy
*cliques*of the*graph*Alt(n + 1, q) of (n + 1)-dimensional alternating forms over GF(q), n ≥ 4, is the unique flag-transitive geometry of rank 3 where ... The elements of Far(D n+1 (q)) of type 2 can be recovered*in*the 0-*graph*as maximal*cliques*of size q 3 . ... they meet*in*a 4-*clique*. ...##
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Pairwise correlation graphs from hippocampal population activity have highly non-random, low-dimensional clique topology

2013
*
BMC Neuroscience
*

We found that pairwise correlation

doi:10.1186/1471-2202-14-s1-p182
pmcid:PMC3704411
fatcat:rolievuzyraadanvaopy2q7aa4
*graphs*are (1) highly non-random,*in*a way that is easily detected by the*clique**topology*; and (2) the*clique**topology*is low-dimensional and consistent with what is ... This can be detected by examining the*clique**topology*of the*graph*(specifically,*topological*invariants called 'Betti numbers' of the*clique*complex), and is closely tied to the dimension and*topology*... We found that pairwise correlation*graphs*are (1) highly non-random,*in*a way that is easily detected by the*clique**topology*; and (2) the*clique**topology*is low-dimensional and consistent with what is ...##
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Exploiting the Formation of Maximal Cliques in Social Networks

2017
*
Symmetry
*

It is believed that our work can provide a new research solution and direction for future

doi:10.3390/sym9070100
fatcat:peljone755chrb46gsbgxmxt6q
*topological*structure analysis*in*various complex networking systems. ... Aiming to exploit the formation of maximal*cliques**in*social networks, this paper pioneers a creative research issue on the detection of bases of maximal*cliques**in*social networks. ... After that,*topological*structure analysis of a social*graph*is formally provided*in*Section 3.2. ...##
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NIT: A New Internet Topology Generator
[chapter]

2009
*
Lecture Notes in Computer Science
*

Internet

doi:10.1007/978-3-642-03700-9_12
fatcat:w675jgktdbcypge27e62cvugyu
*topology*generators play an essential role*in*computer network research. This paper presents a new Internet autonomous system level*topology*generator. ... Synthetic*topologies*generated by the proposed generator are compared to real Internet*topologies*and to synthetic*topologies*generated by a well-known*topology*generator. ... We thank UCLA Internet Research Laboratory for gathering the Internet*topology*data and making it available on the Web. Joylan thanks LABI/UNIOESTE for supporting his work. ...##
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Structural Graph Indexing for Mining Complex Networks

2010
*
2010 IEEE 30th International Conference on Distributed Computing Systems Workshops
*

As indexing feature, we use common

doi:10.1109/icdcsw.2010.42
dblp:conf/icdcsw/KardesG10
fatcat:kek7kwqbazcijhobfk5elefjmi
*graph*structures, namely, star, complete bipartite, triangle and*clique*, that frequently appear*in*protein, chemical compound, and Internet*graphs*. ...*In*many instances, these*graphs*are very large*in*size presenting challenges*in*their analysis. Hence,*graph*indexing techniques are developed to enhance various*graph*mining algorithms. ... EVALUATION ON INTERNET*TOPOLOGIES**In*this section, we use router level Internet*topologies*to analyze the structural*graph*indexing approach. A. ...##
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Graph Planarity by Replacing Cliques with Paths

2020
*
Algorithms
*

Let G be a simple

doi:10.3390/a13080194
fatcat:qm3r3iam5nhwnhiiljafaellva
*topological**graph*whose vertices are partitioned into subsets of size at most h, each inducing a*clique*. h-Clique2Path Planarity asks whether it is possible to obtain a planar subgraph ...*In*particular, we prove that h-Clique2Path Planarity is NP-complete even when h=4 and G is a simple 3-plane*graph*, while it can be solved*in*linear time when G is a simple 1-plane*graph*, for any value ...*In*the version of h-CLIQUE2PATH PLANARITY that we study, the input*graph*G is a simple*topological**graph*. ...##
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Graph theoretic representation of rings of continuous functions

2020
*
Filomat
*

We show that the

doi:10.2298/fil2010417b
fatcat:vf2ow5m4y5ej5fr4kgufydha3i
*graph*is connected and triangulated. We also study the inter-relationship of*cliques*of ?(C(X)) and ideals*in*C(X) which helps to characterize the structure of maximal*cliques*of ? ...*In*this paper, we introduce a*graph*structure, called zero-set intersection*graph*?(C(X)), on the ring of real valued continuous functions, C(X), on a Tychonoff space X. ... We define prime*clique**in*a*graph*to bring the concept of a*clique*to be contained*in*an unique maximal*clique*. 19] . ...##
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jHoles: A Tool for Understanding Biological Complex Networks via Clique Weight Rank Persistent Homology

2014
*
Electronical Notes in Theoretical Computer Science
*

Complex networks equipped with

doi:10.1016/j.entcs.2014.06.011
fatcat:kc3idrvh7vb2pdfpkqdjim2amq
*topological*data analysis are one of the promising tools*in*the study of biological systems (e.g. evolution dynamics, brain correlation, breast cancer diagnosis, etc. . . ... of the filtering process for*clique*weight rank homology. ... We now know that if we want all the*cliques**in*a*graph*, then they are either a subset of a maximal*clique*of the*graph*or a maximal*clique*. ...##
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Impact of Multi-access Links on the Internet Topology Modeling

2013
*
2013 IEEE 21st International Symposium on Modelling, Analysis and Simulation of Computer and Telecommunication Systems
*

*In*this paper,

*in*addition to the thoroughly studied degree distribution, we analyze the subnet and interface distributions of major Internet

*topology*datasets. ... Internet

*topology*measurement studies provide samples of the underlying network at various levels. ... Although reference

*graph*includes up to 20-

*cliques*, 1k or 2k synthetic

*topologies*preserve neither the maximum

*clique*size nor the

*clique*distribution as seen

*in*Figure 2 -a. ...

##
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Turning Cliques into Paths to Achieve Planarity
[article]

2018
*
arXiv
*
pre-print

We study this problem when G is a simple

arXiv:1808.08925v2
fatcat:jw3lxscv3jhndjtj4yjrxqdvzy
*topological**graph*, and establish its complexity*in*relation to k-planarity. ... of size at most h, each inducing a*clique*, remove edges from each*clique*so that the subgraph induced by each subset is a path,*in*such a way that the resulting subgraph of G is planar. ... Since the planarity of a simple*topological**graph*can be checked*in*linear time, the h-C2PP problem for simple*topological*k-plane*graphs*belongs to NP for all values of h and k. ...##
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Clique topology of real symmetric matrices
[article]

2015
*
arXiv
*
pre-print

This is the Supplementary Text for "

arXiv:1502.06173v1
fatcat:k6b6zvcyabbddh7tbqwuputzku
*Clique**topology*reveals intrinsic geometric structure*in*neural correlations," by Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov. ... We refer to these invariants as*clique**topology**in*order to indicate that we are measuring*topological*features of the organization of*cliques**in*the*graph*, rather than the usual*topology*of the*graph*. ...*Clique**topology*across the order complex We now turn our attention to the*clique**topology*of all*graphs**in*the order complex at once. ...##
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Parameterized Verification of Safety Properties in Ad Hoc Network Protocols

2011
*
Electronic Proceedings in Theoretical Computer Science
*

We consider a model

doi:10.4204/eptcs.60.4
fatcat:xgpehk3jg5d33hxp7bwk3nipwm
*in*which the communication*topology*of a network is represented as a*graph*. Nodes represent states of individual processes. Adjacent nodes represent single-hop neighbors. ... least one node*in*a certain state. ... More precisely, we have considered networks*in*which the underlying*topology*is*in*between the class of*cliques*and the strictly larger class of bounded diameter*graphs*. ...
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