IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v36y2018i4d10.1007_s10878-018-0311-9.html
   My bibliography  Save this article

Embeddings into almost self-centered graphs of given radius

Author

Listed:
  • Kexiang Xu

    (Nanjing University of Aeronautics and Astronautics)

  • Haiqiong Liu

    (Nanjing University of Aeronautics and Astronautics)

  • Kinkar Ch. Das

    (Sungkyunkwan University)

  • Sandi Klavžar

    (University of Ljubljana
    University of Maribor
    Institute of Mathematics, Physics and Mechanics)

Abstract

A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius r is called an r-ASC graph. The r-ASC index $$\theta _r(G)$$ θ r ( G ) of a graph G is the minimum number of vertices needed to be added to G such that an r-ASC graph is obtained that contains G as an induced subgraph. It is proved that $$\theta _r(G)\le 2r$$ θ r ( G ) ≤ 2 r holds for any graph G and any $$r\ge 2$$ r ≥ 2 which improves the earlier known bound $$\theta _r(G)\le 2r+1$$ θ r ( G ) ≤ 2 r + 1 . It is further proved that $$\theta _r(G)\le 2r-1$$ θ r ( G ) ≤ 2 r - 1 holds if $$r\ge 3$$ r ≥ 3 and G is of order at least 2. The 3-ASC index of complete graphs is determined. It is proved that $$\theta _3(G)\in \{3,4\}$$ θ 3 ( G ) ∈ { 3 , 4 } if G has diameter 2 and for several classes of graphs of diameter 2 the exact value of the 3-ASC index is obtained. For instance, if a graph G of diameter 2 does not contain a diametrical triple, then $$\theta _3(G) = 4$$ θ 3 ( G ) = 4 . The 3-ASC index of paths of order $$n\ge 1$$ n ≥ 1 , cycles of order $$n\ge 3$$ n ≥ 3 , and trees of order $$n\ge 10$$ n ≥ 10 and diameter $$n-2$$ n - 2 are also determined, respectively, and several open problems proposed.

Suggested Citation

  • Kexiang Xu & Haiqiong Liu & Kinkar Ch. Das & Sandi Klavžar, 2018. "Embeddings into almost self-centered graphs of given radius," Journal of Combinatorial Optimization, Springer, vol. 36(4), pages 1388-1410, November.
  • Handle: RePEc:spr:jcomop:v:36:y:2018:i:4:d:10.1007_s10878-018-0311-9
    DOI: 10.1007/s10878-018-0311-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-018-0311-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10878-018-0311-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kinkar Ch. Das & M. J. Nadjafi-Arani, 2017. "On maximum Wiener index of trees and graphs with given radius," Journal of Combinatorial Optimization, Springer, vol. 34(2), pages 574-587, August.
    2. Guifu Su & Liming Xiong & Xiaofeng Su & Xianglian Chen, 2015. "Some results on the reciprocal sum-degree distance of graphs," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 435-446, October.
    3. Baoyindureng Wu & Xinhui An & Guojie Liu & Guiying Yan & Xiaoping Liu, 2013. "Minimum degree, edge-connectivity and radius," Journal of Combinatorial Optimization, Springer, vol. 26(3), pages 585-591, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Anand, Bijo S. & Chandran S. V., Ullas & Changat, Manoj & Klavžar, Sandi & Thomas, Elias John, 2019. "Characterization of general position sets and its applications to cographs and bipartite graphs," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 84-89.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kinkar Ch. Das & M. J. Nadjafi-Arani, 2017. "On maximum Wiener index of trees and graphs with given radius," Journal of Combinatorial Optimization, Springer, vol. 34(2), pages 574-587, August.
    2. Xiangxiang Liu & Ligong Wang & Xihe Li, 2020. "The Wiener index of hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 351-364, February.
    3. Debarun Ghosh & Ervin Győri & Addisu Paulos & Nika Salia & Oscar Zamora, 2020. "The maximum Wiener index of maximal planar graphs," Journal of Combinatorial Optimization, Springer, vol. 40(4), pages 1121-1135, November.
    4. Leanne Roncolato & John Willoughby, 2017. "Job Quality Complexities," Review of Radical Political Economics, Union for Radical Political Economics, vol. 49(1), pages 30-53, March.
    5. Hongfang Liu & Jinxia Liang & Yuhu Liu & Kinkar Chandra Das, 2023. "A Combinatorial Approach to Study the Nordhaus–Guddum-Type Results for Steiner Degree Distance," Mathematics, MDPI, vol. 11(3), pages 1-19, February.
    6. Siva Venkadesh & Anthony Santarelli & Tyler Boesen & Hong-Wei Dong & Giorgio A. Ascoli, 2023. "Combinatorial quantification of distinct neural projections from retrograde tracing," Nature Communications, Nature, vol. 14(1), pages 1-10, December.
    7. Hamid Darabi & Yaser Alizadeh & Sandi Klavžar & Kinkar Chandra Das, 2021. "On the relation between Wiener index and eccentricity of a graph," Journal of Combinatorial Optimization, Springer, vol. 41(4), pages 817-829, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:36:y:2018:i:4:d:10.1007_s10878-018-0311-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.