Abstract
The \(\varDelta \)-interval of \(u,v \in V (G)\), \(I_{\varDelta }(u,v)\), is the set formed by u, v and every w in V(G) such that \(\{u, v, w\}\) is a triangle \((K_3)\) of G. A set S of vertices such that \(I_{\varDelta } (S)=V(G)\) is called a \(\varDelta \)-set. \(\varDelta \)-number is the minimum cardinality of a \(\varDelta \)-set. \(\varDelta \)-graph is a graph with all the vertices lie on some triangles. If a block graph is a \(\varDelta \)-graph, then we say that it is a block \(\varDelta \)-graph. A set \(S\subseteq V (G)\) is \(\varDelta \)-convex if there is no vertex \(u \in V(G)\setminus S\) forming a triangle with two vertices of S. The convexity number of a graph G with respect to the \(\varDelta \)-convexity is the maximum cardinality of a proper convex subset of G. We have given an exact value for the convexity number of block \(\varDelta \)-graphs with diameter \({\le }3\), block \(\varDelta \)-graphs with diameter \({>}3\) and the two standard graph products (Strong, Lexicographic products), a bound for Cartesian product. Also discussed some bounds for \(\varDelta \)-number and a realization is done for the \(\varDelta \)-number and the hull number.
P. G. Narasimha-Shenoi—Supported by Science and Engineering Research Board, a statutory body of Government of India under their MATRICS Scheme No. MTR/2018/000012.
R. Sabeer Sain—Supported by the University of Kerala for providing University JRF.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Anand, B.S., Anil, A., Changat, M., Dourado, M.C., Sain R.,S.: Computing the hull number in \({\varDelta }\)-convexity (2018, submitted)
Anand, B.S., Changat, M., Klavžar, S., Peterin, I.: Convex sets in lexicographic products of graphs. Graphs Comb. 28(1), 77–84 (2012)
Anand, B.S., Changat, M., Narasimha-Shenoi, P.G.: Helly and exchange numbers of geodesic and Steiner convexities in lexicographic product of graphs. Discrete Math. Algorithms Appl. 7(04), 1550049 (2015)
Anand, B.S., Changat, M., Peterin, I., Narasimha-Shenoi, P.G.: Some Steiner concepts on lexicographic products of graphs. Discrete Math. Algorithms Appl. 6(04), 1450060 (2014)
Changat, M., Mathew, J.: On triangle path convexity in graphs. Discrete Math. 206(1–3), 91–95 (1999)
Changat, M., Mathew, J.: Induced path transit function, monotone and peano axioms. Discrete Math. 286(3), 185–194 (2004)
Changat, M., Mulder, H.M., Sierksma, G.: Convexities related to path properties on graphs. Discrete Math. 290(2–3), 117–131 (2005)
van De Vel, M.L.: Theory of Convex Structures, vol. 50. Elsevier, Burlington (1993)
Dourado, M.C., Penso, L.D., Rautenbach, D.: On the geodetic hull number of \({P}_k\)-free graphs. Theoret. Comput. Sci. 640, 52–60 (2016)
Dourado, M.C., Sampaio, R.M.: Complexity aspects of the triangle path convexity. Discrete Appl. Math. 206, 39–47 (2016)
Duchet, P.: Convexity in combinatorial structures. In: Proceedings of the 14th Winter School on Abstract Analysis, pp. 261–293. Circolo Matematico di Palermo (1987)
Duchet, P.: Convex sets in graphs, II. Minimal path convexity. J. Comb. Theor. Ser. B 44(3), 307–316 (1988)
Farber, M.: Bridged graphs and geodesic convexity. Discrete Math. 66(3), 249–257 (1987)
Jiang, T., Pelayo, I., Pritikin, D.: Geodesic convexity and Cartesian products in graphs. Graphs Comb. (2004, submitted)
Kannan, B., Changat, M.: Hull numbers of path convexities on graphs. In: Proceedings of the International Instructional Workshop on Convexity in Discrete Structures, vol. 5, pp. 11–23 (2008)
Morgana, M.A., Mulder, H.M.: The induced path convexity, betweenness, and svelte graphs. Discrete Math. 254(1–3), 349–370 (2002)
Mulder, H.M.: The Interval Function of a Graph. MC Tracts. Mathematisch Centrum, Amsterdam (1980)
Mulder, M.H.: Transit functions on graphs (and posets). Technical report (2007)
Pelayo, I.M.: Geodesic Convexity in Graphs. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-8699-2
Polat, N., Sabidussi, G.: On the geodesic pre-hull number of a graph. Eur. J. Comb. 30(5), 1205–1220 (2009)
Sierksma, G.: Convexity on unions of sets. Compositio Mathematica 42(3), 391–400 (1980)
Acknowledgements
We are grateful to the anonymous referees for their numerous valuable comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Anand, B.S., Narasimha-Shenoi, P.G., Ramla, S.S. (2020). \(\varDelta \)-Convexity Number and \(\varDelta \)-Number of Graphs and Graph Products. In: Changat, M., Das, S. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2020. Lecture Notes in Computer Science(), vol 12016. Springer, Cham. https://doi.org/10.1007/978-3-030-39219-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-39219-2_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-39218-5
Online ISBN: 978-3-030-39219-2
eBook Packages: Computer ScienceComputer Science (R0)