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

Rebecca Robinson, Graham Farr
2017 arXiv   pre-print
The 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  ... 
arXiv:1705.01224v2 fatcat:ta74d3t22bajxlwdzxl6mdipla

Extremal connectivity for topological cliques in bipartite graphs

Daniela Kühn, Deryk Osthus
2006 Journal of combinatorial theory. Series B (Print)  
Given an ε-regular bipartite 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.  ... 
doi:10.1016/j.jctb.2005.06.005 fatcat:uwjpfecwlvemdb4c3rtum7yq5u

Large Topological Cliques in Graphs Without a 4-Cycle

DANIELA KHN, DERYK OSTHUS
2004 Combinatorics, probability & computing  
Mader asked whether every C 4 -free 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] .  ... 
doi:10.1017/s0963548303005650 fatcat:akwy2bpll5drbdkadaj7obshki

A characterization of the geometry of large maximal cliques of the alternating forms graph

Antonio Pasini
2008 Innovations in Incidence Geometry Algebraic Topological and Combinatorial  
We prove that the geometry of vertices, edges and q n -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.  ... 
doi:10.2140/iig.2008.8.81 fatcat:zbgienbtbnf4janpltb23xwwpy

Pairwise correlation graphs from hippocampal population activity have highly non-random, low-dimensional clique topology

Carina Curto, Chad Giusti, Keler Marku, Eva Pastalkova, Vladimir Itskov
2013 BMC Neuroscience  
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  ...  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  ... 
doi:10.1186/1471-2202-14-s1-p182 pmcid:PMC3704411 fatcat:rolievuzyraadanvaopy2q7aa4

Exploiting the Formation of Maximal Cliques in Social Networks

Fei Hao, Doo-Soon Park, Zheng Pei
2017 Symmetry  
It is believed that our work can provide a new research solution and direction for future 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.  ... 
doi:10.3390/sym9070100 fatcat:peljone755chrb46gsbgxmxt6q

NIT: A New Internet Topology Generator [chapter]

Joylan Nunes Maciel, Cristina Duarte Murta
2009 Lecture Notes in Computer Science  
Internet 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.  ... 
doi:10.1007/978-3-642-03700-9_12 fatcat:w675jgktdbcypge27e62cvugyu

Structural Graph Indexing for Mining Complex Networks

Hakan Kardes, Mehmet Hadi Gunes
2010 2010 IEEE 30th International Conference on Distributed Computing Systems Workshops  
As indexing feature, we use common 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.  ... 
doi:10.1109/icdcsw.2010.42 dblp:conf/icdcsw/KardesG10 fatcat:kek7kwqbazcijhobfk5elefjmi

Graph Planarity by Replacing Cliques with Paths

Patrizio Angelini, Peter Eades, Seok-Hee Hong, Karsten Klein, Stephen Kobourov, Giuseppe Liotta, Alfredo Navarra, Alessandra Tappini
2020 Algorithms  
Let G be a simple 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.  ... 
doi:10.3390/a13080194 fatcat:qm3r3iam5nhwnhiiljafaellva

Graph theoretic representation of rings of continuous functions

Bedanta Bose, Angsuman Das
2020 Filomat  
We show that the 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] .  ... 
doi:10.2298/fil2010417b fatcat:vf2ow5m4y5ej5fr4kgufydha3i

jHoles: A Tool for Understanding Biological Complex Networks via Clique Weight Rank Persistent Homology

Jacopo Binchi, Emanuela Merelli, Matteo Rucco, Giovanni Petri, Francesco Vaccarino
2014 Electronical Notes in Theoretical Computer Science  
Complex networks equipped with 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.  ... 
doi:10.1016/j.entcs.2014.06.011 fatcat:kc3idrvh7vb2pdfpkqdjim2amq

Impact of Multi-access Links on the Internet Topology Modeling

Mehmet Burak Akgun, Mehmet Hadi Gunes
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.  ... 
doi:10.1109/mascots.2013.60 dblp:conf/mascots/AkgunG13 fatcat:6rgddrtj3ncxdeviqnxkar3tn4

Turning Cliques into Paths to Achieve Planarity [article]

Patrizio Angelini, Peter Eades, Seok-Hee Hong, Karsten Klein, Stephen Kobourov, Giuseppe Liotta, Alfredo Navarra, Alessandra Tappini
2018 arXiv   pre-print
We study this problem when G is a simple 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.  ... 
arXiv:1808.08925v2 fatcat:jw3lxscv3jhndjtj4yjrxqdvzy

Clique topology of real symmetric matrices [article]

Carina Curto, Chad Giusti, Vladimir Itskov
2015 arXiv   pre-print
This is the Supplementary Text for "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.  ... 
arXiv:1502.06173v1 fatcat:k6b6zvcyabbddh7tbqwuputzku

Parameterized Verification of Safety Properties in Ad Hoc Network Protocols

Giorgio Delzanno, Arnaud Sangnier, Gianluigi Zavattaro
2011 Electronic Proceedings in Theoretical Computer Science  
We consider a model 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.  ... 
doi:10.4204/eptcs.60.4 fatcat:xgpehk3jg5d33hxp7bwk3nipwm
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