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Perfect Italian domination in cographs

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  • Banerjee, S.
  • Henning, Michael A.
  • Pradhan, D.

Abstract

For a graph G=(VG,EG), a perfect Italian dominating function on G is a function g: VG → {0, 1, 2} satisfying the condition that for each vertex v with g(v)=0, the sum of the function values assigned to the neighbors of v is exactly two, that is, ∑g(u)=2 where the sum is taken over all neighbors of v. The weight of g, denoted by w(g) is defined ∑g(v) where the sum is taken over all v ∈ VG. The perfect Italian domination number of G, denoted γIp(G), is the minimum weight of a perfect Italian dominating function of G. In this paper, we prove that the perfect Italian domination number of a connected cograph, a graph containing no induced path on four vertices, belongs to {1, 2, 3, 4} or equals to the order of the cograph. We prove that there is no connected cograph with perfect Italian domination number k, where k ∈ {5, 6, 7, 8, 9}. We also show that for any positive integer k, k ∉ {5, 6, 7, 8, 9}, there exists a connected cograph whose perfect Italian domination number is k. Moreover, we devise a linear time algorithm that computes the perfect Italian domination number in cographs.

Suggested Citation

  • Banerjee, S. & Henning, Michael A. & Pradhan, D., 2021. "Perfect Italian domination in cographs," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    • Handle: RePEc:eee:apmaco:v:391:y:2021:i:c:s0096300320306561
    DOI: 10.1016/j.amc.2020.125703
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    References listed on IDEAS

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    1. H. Abdollahzadeh Ahangar & Michael A. Henning & Christian Löwenstein & Yancai Zhao & Vladimir Samodivkin, 2014. "Signed Roman domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 241-255, February.
    2. Yue, Jun & Wei, Meiqin & Li, Min & Liu, Guodong, 2018. "On the double Roman domination of graphs," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 669-675.
    3. Yongtang Shi & Meiqin Wei & Jun Yue & Yan Zhao, 2017. "Coupon coloring of some special graphs," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 156-164, January.
    4. Chen, He & Jin, Zemin, 2017. "Coupon coloring of cographs," Applied Mathematics and Computation, Elsevier, vol. 308(C), pages 90-95.
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