Abstract
A subset of vertices in a graph G is called a maximum dissociation set if it induces a subgraph with vertex degree at most 1 and the subset has maximum cardinality. The dissociation number of G, denoted by \(\psi (G)\), is the cardinality of a maximum dissociation set. A subcubic tree is a tree of maximum degree at most 3. In this paper, we give the lower and upper bounds on the dissociation number in a subcubic tree of order n and show that the number of maximum dissociation sets of a subcubic tree of order n and dissociation number \(\psi \) is at most \(1.466^{4n-5\psi +2}\).
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Acharya HB, Choi T, Bazzi RA, Gouda MG (2012) The \(k\)-observer problem in computer networks. Networking Sci. 1:15–22
Alekseev VE, Boliac R, Korobitsyn DV, Lozin VV (2007) NP-hard graph problems and boundary classes of graphs. Theoret. Comput. Sci. 389(12):219–236
Alvarado JD, Dantas S, Mohr E, Rautenbach D (2019) On the maximum number of minimum dominating sets in forests. Discrete Math. 342:934–942
Bondy JA, Murty USR (2008) Graph Theory. Springer, New York
Brešar B, Kardo F, Katreni J, Semaniin G (2011) Minimum \(k\)-path vertex cover. Discrete Appl. Math. 159(12):1189–1195
Bród D, Skupień Z (2006) Trees with extremal numbers of dominating sets. Australas. J. Combin. 35:273–290
Cameron K, Hell P (2006) Independent packings in structured graphs. Math. Program. 105(2–3):201–213
Connolly S, Gabor Z, Godbole A, Kay B, Kelly T (2016) Bounds on the maximum number of minimum dominating sets. Discrete Math. 339:1537–1542
Conte A, Marino A, Grossi R, Uno T, Versari L (2022) Proximity search for maximal subgraph enumeration. SIAM J. Comput. 51:1580–1625
Derikvand T, Oboudi MR (2014) On the number of maximum independent sets of graphs. Transactions on Combin. 3(1):29–36
Fomin FV, Grandoni F, Pyatkin AV, Stepanov AA (2005) Bounding the number of minimal dominating sets: a measure and conquer approach. Lecture Notes in Comput. Sci. 3827:573–582
Füredi Z (1987) The number of maximal independent sets in connected graphs. J. Graph Theory 11:463–470
Griggs JR, Grinstead CM, Guichard DR (1988) The number of maximal independent sets in a connected graph. Discrete Math. 68:211–220
Havet F, Kang RJ, Sereni JS (2009) Improper colouring of unit disk graphs. Networks. 54:150–164
Jou MJ, Chang GJ (2000) The number of maximum independent sets in graphs. Taiwan. J. Math. 4:685–695
Kardoš F, Katrenič J, Schiermeyer I (2011) On computing the minium 3-path vertex cover and dissociation number of graphs. Theoret. Comput. Sci. 412:7009–7017
Katrenič J (2016) A faster FPT algorithm for 3-path vertex cover. Inform. Process. Lett. 116:273–278
Koh KM, Goh CY, Dong FM (2008) The maximum number of maximal independent sets in unicyclic connected graphs. Discrete Math. 308:3761–3769
Liu J (1993) Maximal independent sets in bipartite graphs. J. Graph Theory 17(4):495–507
Mohr E, Rautenbach D (2018) On the maximum number of maximum independent sets. Graphs Combin. 34:1729–1740
Mohr E, Rautenbach D (2021) On the maximum number of maximum independent sets in connected graphs. J. Graph Theory 96:510–521
Moon JW, Moser L (1965) On cliques in graphs. Israel J. Math. 3:23–28
Orlovich Y, Dolguib A, Finkec G, Gordond V, Wernere F (2011) The complexity of dissociation set problems in graphs. Discrete Appl. Math. 159(13):1352–1366
Sagan BE (1988) A note on independent sets in trees. SIAM J. Discrete Math. 1:105–108
Sagan BE, Vatter VR (2006) Maximal and maximum independent sets in graphs with at most \(r\) cycles. J. Graph Theory 53:283–314
Tsukiyama S, Ide M, Ariyoshi H, Shirakawa I (1977) A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6:505–517
Tu JH, Zhou WL (2011) A primal-dual approximation algorithm for the vertex cover \(P_3\) problem. Theoret. Comput. Sci. 412:7044–7048
Tu JH, Zhang ZP, Shi YT (2021) The maximum number of maximum dissociation sets in trees. J. Graph Theory 96:472–489
Wilf HS (1986) The number of maximal independent sets in a tree. SIAM J. Alg. Discrete Methods 7:125–130
Xiao M, Kou S (2017) Exact algorithms for the maximum dissociation set and minimum 3-path vertex cover problems. Theoret. Comput. Sci. 657:86–97
Yannakakis M (1981) Node-deletion problems on bipartite graphs. SIAM J. Comput. 10:310–327
Zito J (1991) The structure and maximum number of maximum independent sets in trees. J. Graph Theory 15:207–221
Funding
This work was supported by Beijing Natural Science Foundation (No. 1232005), Research Foundation for Advanced Talents of Beijing Technology and Business University (No. 19008023211), National Key Research and Development Program of China (2019YFC1906102), and National Key Technology Research and Development Program of the Ministry of Science and Technology of China (No. 2015BAK39B00).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, L., Tu, J. & Xin, C. Maximum dissociation sets in subcubic trees. J Comb Optim 46, 8 (2023). https://doi.org/10.1007/s10878-023-01076-9
Accepted:
Published:
DOI: https://doi.org/10.1007/s10878-023-01076-9