Skip to main content
SpringerLink
Log in

Spanning Trees with Few Branch Vertices in Graphs of Bounded Neighborhood Diversity

  • Conference paper
  • First Online:
Structural Information and Communication Complexity (SIROCCO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13892))

  • 324 Accesses

Abstract

A branch vertex in a tree is a vertex of degree at least three. We study the NP-hard problem of constructing spanning trees with as few branch vertices as possible. This problem generalizes the famous Hamiltonian Path problem which corresponds to the case of no vertices having degree three or more. It has been extensively studied in the literature and has important applications in network design and optimization. In this paper, we study the problem of finding a spanning tree with the minimum number of branch vertices in graphs of bounded neighborhood diversity. Neighborhood diversity, a generalization of vertex cover to dense graphs, plays an important role in the design of algorithms for such graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In the following we use omit the graph name G (e.g. we use d(v), instead of \(d_G(v)\)) whenever the graph is clear from the context.

References

  1. Agrawal, A., et al.: Parameterized complexity of happy coloring problems. Theoret. Comput. Sci. 835, 58–81 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araujo, J., Ducoffe, G., Nisse, N., Suchan, K.: On interval number in cycle convexity. Discrete Math. Theoret. Comput. Sci. 20(1), 1–28 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Watel, D., Baste, J.: An FPT algorithm for node-disjoint subtrees problems parameterized by treewidth. Available at SSRN: https://ssrn.com/abstract=4197048 or https://dx.doi.org/10.2139/ssrn.4197048

  4. Bermond, J.-C., Gargano, L., Rescigno, A.A.: Gathering with minimum delay in tree sensor networks. In: Shvartsman, A.A., Felber, P. (eds.) SIROCCO 2008. LNCS, vol. 5058, pp. 262–276. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-69355-0_22

    Chapter  Google Scholar 

  5. Bermond, J.-C., Gargano, L., Rescigno, A.A.: Gathering with minimum completion time in sensor tree networks. J. Interconnect. Netw. 11, 1–33 (2010)

    Article  Google Scholar 

  6. Bermond, J.-C., Gargano, L., Peénnes, S., Rescigno, A.A., Vaccaro, U.: Optimal time data gathering in wireless networks with multidirectional antennas. Theoret. Comput. Sci. 509, 122–139 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhyravarapu, S., Reddy, I.V.: On structural parameterizations of star coloring. arXiv preprint: arXiv:2211.12226 (2022)

  8. Bianzino, A.P., Chaudet, C., Rossi, D., Rougier, J.L.: A survey of green networking research. IEEE Commun. Surv. Tutorials 14, 3–20 (2012)

    Article  Google Scholar 

  9. Carrabs, F., Cerulli, R., Gaudioso, M., Gentili, M.: Lower and upper bounds for the spanning tree with minimum branch vertices. Comput. Optim. Appl. 56(2), 405–438 (2013). https://doi.org/10.1007/s10589-013-9556-5

    Article  MathSciNet  MATH  Google Scholar 

  10. Celik, A., Kamal, A.E.: Green cooperative spectrum sensing and scheduling in heterogeneous cognitive radio networks. IEEE Trans. Cognitive Commun. Networking 2, 238–248 (2016)

    Article  Google Scholar 

  11. Cerrone, C., Cerulli, R., Raiconi, A.: Relations, models and a memetic approach for three degree-dependent spanning tree problems. Eur. J. Oper. Res. 232(3), 442–453 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cerulli, R., Gentili, M., Iossa, A.: Bounded-degree spanning tree problems: models and new algorithms. Comput. Optim. Appl. 42(3), 353–370 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chimani, V.M., Spoerhase, J.: Approximating spanning trees with few branches. Theory Comput. Syst. 56(1), 181–196 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cordasco, G., Gargano, L., Rescigno, A.A., Vaccaro, U.: Evangelism in social networks: algorithms and complexity. Networks 71(4), 346–357 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cordasco, G., Gargano, L., Rescigno, A.A.: Iterated type partitions. In: Gąsieniec, L., Klasing, R., Radzik, T. (eds.) IWOCA 2020. LNCS, vol. 12126, pp. 195–210. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48966-3_15

    Chapter  Google Scholar 

  16. Cordasco, G., Gargano, L., Rescigno, A.A.: Parameterized complexity of immunization in the threshold model. In: Mutzel, P., Rahman, M.S., Slamin (eds.) WALCOM: Algorithms and Computation. WALCOM 2022. Lecture Notes in Computer Science(), vol. 13174. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-96731-4_23

  17. Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discret. Appl. Math. 101(1–3), 77–144 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cygan, M., et al.: Lower bounds for kernelization. In: Parameterized Algorithms, pp. 523–555. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3_15

    Chapter  Google Scholar 

  19. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer-Verlag, New York (1999)

    Book  MATH  Google Scholar 

  20. Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Clique-width: on the price of generality. In: Proceedings of SODA 2009, pp. 825–834 (2009)

    Google Scholar 

  21. Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Intractability of clique-width parameterizations. SIAM J. Comput. 39(5), 1941–1956 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gajarsky, J., Lampis, M., Ordyniak, S.: Parameterized algorithms for modular-width. In: Gutin, G., Szeider, S. (eds.) Parameterized and Exact Computation. IPEC 2013. Lecture Notes in Computer Science, vol. 8246. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-03898-8_15

  23. Ganian, R.: Using neighborhood diversity to solve hard problems (2012). arXiv:1201.3091

  24. Gargano, L., Hammar, M., Hell, P., Stacho, L., Vaccaro, U.: Spanning spiders and light splitting switches. Discret. Math. 285(1), 83–95 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gargano, L., Hell, P., Stacho, L., Vaccaro, U.: Spanning trees with bounded number of branch vertices. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 355–365. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45465-9_31

    Chapter  Google Scholar 

  26. Gargano, L., Rescigno, A.A.: Complexity of conflict-free colorings of graphs. Theoret. Comput. Sci. 566, 39–49 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gargano, L., Rescigno, A.A.: Collision-free path coloring with application to minimum-delay gathering in sensor networks. Discret. Appl. Math. 157(8), 1858–1872 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gavenciak, T., Koutecký, M., Knop, D.: Integer programming in parameterized complexity: five miniatures. Discrete Optim. 44, 100596 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gozupek, D., Buhari, S., Alagoz, F.: A spectrum switching delay-aware scheduling algorithm for centralized cognitive radio networks. IEEE Trans. Mobile Comp. 12, 1270–1280 (2013)

    Article  Google Scholar 

  30. Jansen, K., Rohwedderb, L.: On integer programming, discrepancy, and convolution. Math. Operat. Res., 1–15 (2023)

    Google Scholar 

  31. Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64, 19–37 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Landete, M., Marín, A., Sainz-Pardo, J.L.: Decomposition methods based on articulation vertices for degree-dependent spanning tree problems. Comput. Optim. Appl. 68(3), 749–773 (2017). https://doi.org/10.1007/s10589-017-9924-7

    Article  MathSciNet  MATH  Google Scholar 

  33. Marin, A.: Exact and heuristic solutions for the minimum number of branch vertices spanning tree problem. Eur. J. Oper. Res. 245(3), 680–689 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Melo, R.A., Samer, P., Urrutia, S.: An effective decomposition approach and heuristics to generate spanning trees with a small number of branch vertices. Comput. Optim. Appl. 65(3), 821–844 (2016). https://doi.org/10.1007/s10589-016-9850-0

    Article  MathSciNet  MATH  Google Scholar 

  35. Matsuda, H., Ozeki, K., Yamashita, T.: Spanning trees with a bounded number of branch vertices in a claw-free graph. Graphs and Combinatorics 30, 429–437 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Robertson, N., Seymour, P.D.: Graph minors II. algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rossi, A., Singh, A., Shyam, S.: Cutting-plane-based algorithms for two branch vertices related spanning tree problems. Optim. Eng. 15, 855–887 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Salamon, G.: Spanning tree optimization problems with degree-based objective functions. In: 4th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, pp. 309–315 (2005)

    Google Scholar 

  39. Shami, N., Rasti, M.: A joint multi-channel assignment and power control scheme for energy efficiency in cognitive radio networks. In: Proceedings IEEE Wireless Communications and Networking Conference, WCNC 2016, pp. 1–6 (2016)

    Google Scholar 

  40. Silva, R.M.A., Silva, D.M., Resende, M.G.C., Mateus, G.R., Goncalves, J.F., Festa, P.: An edge-swap heuristic for generating spanning trees with minimum number of branch vertices. Optim. Lett. 8(4), 1225–1243 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Silvestri, S., Laporte, G., Cerulli, R.: A branch-and-cut algorithm for the minimum branch vertices spanning tree problem. Comput. Oper. Res. 81, 322–332 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Sundar, S., Singh, A., Rossi, A.: New heuristics for two bounded-degree spanning tree problems. Inf. Sci. 195, 226–240 (2012)

    Article  MathSciNet  Google Scholar 

  43. Toufar, T., Masařík, T., Koutecký, M., Knop, D.: Simplified algorithmic metatheorems beyond MSO: treewidth and neighborhood diversity. Log. Methods Comput. Sci. 15 (2019)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Salerno, Fisciano, Italy

    Luisa Gargano & Adele A. Rescigno

Authors
  1. Luisa Gargano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adele A. Rescigno
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adele A. Rescigno .

Editor information

Editors and Affiliations

  1. National Autonomous University of Mexico, Mexico, Mexico

    Sergio Rajsbaum

  2. Gran Sasso Science Institute, L’Aquila, Italy

    Alkida Balliu

  3. Arizona State University, Tempe, AZ, USA

    Joshua J. Daymude

  4. Gran Sasso Science Institute, L’Aquila, Italy

    Dennis Olivetti

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gargano, L., Rescigno, A.A. (2023). Spanning Trees with Few Branch Vertices in Graphs of Bounded Neighborhood Diversity. In: Rajsbaum, S., Balliu, A., Daymude, J.J., Olivetti, D. (eds) Structural Information and Communication Complexity. SIROCCO 2023. Lecture Notes in Computer Science, vol 13892. Springer, Cham. https://doi.org/10.1007/978-3-031-32733-9_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-32733-9_22

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-32732-2

  • Online ISBN: 978-3-031-32733-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation