Abstract
A graph G = (V,E) is γ-quasi-complete (γ ∈ [0,1]) if every vertex in G is connected to at least γ.(|V| − 1) other vertices. In this paper, we establish some relationships between the girth and the quasi-completeness of a graph. We also derive an upper bound \(\frac{1}{2}\big(1+\frac{r}{\gamma}\big) + \sqrt{\frac{1}{4}\big(1+\frac{r}{\gamma}\big)^2 + \frac{2|E|}{\gamma} - \frac{r|V|}{\gamma}}\) for the largest order γ-quasi-complete subgraph in a graph of minimum degree r.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abello, J., Resende, M.G.C., Sudarsky, S.: Massive Quasi-Clique Detection. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 598–612. Springer, Heidelberg (2002)
Amin, A.T., Hakimi, S.L.: Upper bounds on the order of a clique of a graph. SIAM J. Appl. Math. 22, 569–573 (1972)
Bandyopadhyay, S., Bhattacharyya, M.: Mining the Largest Dense Vertexlet in a Weighted Scale-free Graph. Fund. Inform. 96(1-2), 1–25 (2009)
Bhattacharyya, M., Bandyopadhyay, S.: Analyzing Topological Properties of Protein-protein Interaction Networks: A Perspective towards Systems Biology. In: Computational Intelligence and Pattern Analysis in Biological Informatics, pp. 349–368. John Wiley & Sons, Inc., Hoboken (2010)
Bollobás, B.: Modern Graph Theory. Graduate Texts in Mathematics, vol. 184. Springer, New York (1998)
Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The Maximum Clique Problem. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization: Supplementary Volume A, pp. 1–74. Kluwer Academic, Dordrecht (1999)
Brakerski, Z., Patt-Shamir, B.: Distributed discovery of large near-cliques In: Proceedings of the 23rd International Conference on Distributed Computing, Elche, Spain, pp. 206–220 (2009)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2009)
Jiang, D., Pei, J.: Mining Frequent Cross-Graph Quasi-Cliques. ACM Trans. Knowl. Discov. Data 2(4), 16 (2009)
Pei, J., Jiang, D., Zhang, A.: On Mining Cross-Graph Quasi-Cliques. In: Proceedings of the 11th ACM SIGKDD, Chicago, Illinois, USA, pp. 228–238 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bhattacharyya, M., Bandyopadhyay, S. (2012). Bounds on Quasi-Completeness. In: Arumugam, S., Smyth, W.F. (eds) Combinatorial Algorithms. IWOCA 2012. Lecture Notes in Computer Science, vol 7643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35926-2_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-35926-2_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35925-5
Online ISBN: 978-3-642-35926-2
eBook Packages: Computer ScienceComputer Science (R0)