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Roman [1,2]-domination of graphs

Author

Listed:
  • Hao, Guoliang
  • Chen, Xiaodan
  • Sheikholeslami, Seyed Mahmoud
  • Jiang, Haining

Abstract

A [1,2]-set of a graph G is a set S of vertices of G if every vertex not in S has one or two neighbors in S. The [1,2]-domination number γ[1,2](G) of G equals the minimum cardinality of a [1,2]-set of G. A Roman [1,2]-dominating function (R[1,2]DF) on a graph G is a function f from the vertex set V of G to the set {0,1,2} such that any vertex assigned 0 under f has one or two neighbors assigned 2. The weight of an R[1,2]DF f is the sum ∑x∈Vf(x). The Roman [1,2]-domination number γR[1,2](G) of G equals the minimum weight of an R[1,2]DF on G. In this paper, we prove that the decision problem on the Roman [1,2]-domination is NP-complete for bipartite and chordal graphs. Moreover, we give some bounds on the Roman [1,2]-domination number. In particular, we show that for any nontrivial tree T, γR[1,2](T)≥γ[1,2](T)+1 and characterize all trees obtaining equality in this bound.

Suggested Citation

  • Hao, Guoliang & Chen, Xiaodan & Sheikholeslami, Seyed Mahmoud & Jiang, Haining, 2024. "Roman [1,2]-domination of graphs," Applied Mathematics and Computation, Elsevier, vol. 468(C).
    • Handle: RePEc:eee:apmaco:v:468:y:2024:i:c:s0096300323006896
    DOI: 10.1016/j.amc.2023.128520
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