Opuscula Math. 41, no. 2 (2021), 259-268
https://doi.org/10.7494/OpMath.2021.41.2.259
Opuscula Mathematica
Remarks on the outer-independent double Italian domination number
Abstract. Let \(G\) be a graph with vertex set \(V(G)\). If \(u\in V(G)\), then \(N[u]\) is the closed neighborhood of \(u\). An outer-independent double Italian dominating function (OIDIDF) on a graph \(G\) is a function \(f:V(G)\longrightarrow \{0,1,2,3\}\) such that if \(f(v)\in\{0,1\}\) for a vertex \(v\in V(G)\), then \(\sum_{x\in N[v]}f(x)\ge 3\), and the set \(\{u\in V(G):f(u)=0\}\) is independent. The weight of an OIDIDF \(f\) is the sum \(\sum_{v\in V(G)}f(v)\). The outer-independent double Italian domination number \(\gamma_{oidI}(G)\) equals the minimum weight of an OIDIDF on \(G\). In this paper we present Nordhaus-Gaddum type bounds on the outer-independent double Italian domination number which improved corresponding results given in [F. Azvin, N. Jafari Rad, L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. 6 (2021), 123-136]. Furthermore, we determine the outer-independent double Italian domination number of some families of graphs.
Keywords: double Italian domination number, outer-independent double Italian domination number, Nordhaus-Gaddum bound.
Mathematics Subject Classification: 05C69.
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- Lutz Volkmann
- RWTH Aachen University, Lehrstuhl II für Mathematik, 52056 Aachen, Germany
- Communicated by Dalibor Fronček.
- Received: 2020-11-09.
- Revised: 2021-01-31.
- Accepted: 2021-02-09.
- Published online: 2021-03-17.
