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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
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    Citations

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    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. 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.
    7. 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.
    8. 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.

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