IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v26y2013i3d10.1007_s10878-012-9482-y.html
   My bibliography  Save this article

Roman domination on strongly chordal graphs

Author

Listed:
  • Chun-Hung Liu

    (National Taiwan University
    Georgia Institute of Technology)

  • Gerard J. Chang

    (National Taiwan University
    National Taiwan University
    Taipei Office)

Abstract

Given real numbers b≥a>0, an (a,b)-Roman dominating function of a graph G=(V,E) is a function f:V→{0,a,b} such that every vertex v with f(v)=0 has a neighbor u with f(u)=b. An independent/connected/total (a,b)-Roman dominating function is an (a,b)-Roman dominating function f such that {v∈V:f(v)≠0} induces a subgraph without edges/that is connected/without isolated vertices. For a weight function $w{:} V\to\Bbb{R}$ , the weight of f is w(f)=∑ v∈V w(v)f(v). The weighted (a,b)-Roman domination number $\gamma^{(a,b)}_{R}(G,w)$ is the minimum weight of an (a,b)-Roman dominating function of G. Similarly, we can define the weighted independent (a,b)-Roman domination number $\gamma^{(a,b)}_{Ri}(G,w)$ . In this paper, we first prove that for any fixed (a,b) the (a,b)-Roman domination and the total/connected/independent (a,b)-Roman domination problems are NP-complete for bipartite graphs. We also show that for any fixed (a,b) the (a,b)-Roman domination and the total/connected/weighted independent (a,b)-Roman domination problems are NP-complete for chordal graphs. We then give linear-time algorithms for the weighted (a,b)-Roman domination problem with b≥a>0, and the weighted independent (a,b)-Roman domination problem with 2a≥b≥a>0 on strongly chordal graphs with a strong elimination ordering provided.

Suggested Citation

  • Chun-Hung Liu & Gerard J. Chang, 2013. "Roman domination on strongly chordal graphs," Journal of Combinatorial Optimization, Springer, vol. 26(3), pages 608-619, October.
    • Handle: RePEc:spr:jcomop:v:26:y:2013:i:3:d:10.1007_s10878-012-9482-y
    DOI: 10.1007/s10878-012-9482-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-012-9482-y
    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/s10878-012-9482-y?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.

    Citations

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


    Cited by:

    1. Abel Cabrera Martínez & Suitberto Cabrera García & Andrés Carrión García & Frank A. Hernández Mira, 2020. "Total Roman Domination Number of Rooted Product Graphs," Mathematics, MDPI, vol. 8(10), pages 1-13, October.
    2. Tian Liu & Chaoyi Wang & Ke Xu, 2015. "Large hypertree width for sparse random hypergraphs," Journal of Combinatorial Optimization, Springer, vol. 29(3), pages 531-540, April.
    3. Abel Cabrera Martínez & Suitberto Cabrera García & Andrés Carrión García, 2020. "Further Results on the Total Roman Domination in Graphs," Mathematics, MDPI, vol. 8(3), pages 1-8, March.
    4. J. Amjadi & S. M. Sheikholeslami & M. Soroudi, 2018. "Nordhaus–Gaddum bounds for total Roman domination," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 126-133, January.
    5. Abel Cabrera Martínez & Dorota Kuziak & Iztok Peterin & Ismael G. Yero, 2020. "Dominating the Direct Product of Two Graphs through Total Roman Strategies," Mathematics, MDPI, vol. 8(9), pages 1-13, August.
    6. Ahlam Almulhim & Bana Al Subaiei & Saiful Rahman Mondal, 2024. "Survey on Roman {2}-Domination," Mathematics, MDPI, vol. 12(17), pages 1-20, September.
    7. Cai-Xia Wang & Yu Yang & Hong-Juan Wang & Shou-Jun Xu, 2021. "Roman {k}-domination in trees and complexity results for some classes of graphs," Journal of Combinatorial Optimization, Springer, vol. 42(1), pages 174-186, July.
    8. Abolfazl Poureidi & Nader Jafari Rad, 2020. "Algorithmic and complexity aspects of problems related to total Roman domination for graphs," Journal of Combinatorial Optimization, Springer, vol. 39(3), pages 747-763, April.
    9. Abel Cabrera Martínez & Juan C. Hernández-Gómez & José M. Sigarreta, 2021. "On the Quasi-Total Roman Domination Number of Graphs," Mathematics, MDPI, vol. 9(21), pages 1-11, November.

    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:26:y:2013:i:3:d:10.1007_s10878-012-9482-y. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.