Roman [1,2]-domination of graphs

Guoliang Hao, Xiaodan Chen, Seyed Mahmoud Sheikholeslami and Haining Jiang

Applied Mathematics and Computation, 2024, vol. 468, issue C

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.

Keywords: Roman [1,2]-domination; [1,2]-domination; Roman domination; NP-complete (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:468:y:2024:i:c:s0096300323006896

DOI: 10.1016/j.amc.2023.128520

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