Restrained Roman and restrained Italian domatic numbers of graphs (Q2081477)

Let \(G=(V,E)\) be a simple graph. A dominating set of \(G\) is a subset \(D\subseteq V\) such that every vertex not in \(D\) is adjacent to at least one vertex in \(D\). The cardinality of the smallest dominating set of \(G\), denoted by \(\gamma(G)\), is the domination number of \(G\). A Roman dominating function (RDF) on \(G\) is defined as a function \(f:V(G)\rightarrow \{0,1,2\}\) satisfying the condition that each vertex \(u\) with \(f(u)=0\) has a neighbor \(v\) with \(f(v)=2\). The weight of an RDF \(f\) is the value \(f(V(G))=\sum_{u\in V(G)} f (u)\). The Roman domination number \(\gamma_R(G)\) is the minimum weight of an RDF on \(G\). An Italian dominating function (IDF) is a function \(f:V(G)\rightarrow \{0,1,2\}\) having the property that \(f(N(u))\geq 2\) for every vertex \(u\) with \(f(u)=0\). The weight of an IDF \(f\) is the value \(f(V(G))=\sum_{u\in V(G)} f (u)\). The Italian domination number \(\gamma_I(G)\) is the minimum weight of an IDF on \(G\). An RDF or IDF \(f\) can be represented by the ordered partition \((V_0,V_1,V_2)\) of \(G\), where \(V_i =\{v \in V (G): f (v) = i\}\) for \(i\in\{0,1,2\}\). A set \(\{f_1,f_2,\dots,f_d\}\) of distinct Roman (Italian) dominating function on \(G\) with the property that \(\sum_{i=1}^d f_i(v)\leq 2\) for each \(v\in V(G)\) is called a Roman (Italian) dominating family (of functions) on \(G\). The maximum number of functions in a Roman (Italian) dominating family on \(G\) is the Roman (Italian) domatic number of \(G\), denoted by \(d_R(G)\) \((d_I(G))\). The restrained Italian dominating function (RIDF) is an Italian dominating function \(f\) with the property that the subgraph induced by \(V_0\) does not have an isolated vertex. The restrained Italian domination number \(\gamma_{rI}(G)\) equals the minimum weight of an RIDF on \(G\). A set \(\{f_1,f_2,\dots,f_d\}\) of distinct restrained Roman (Italian) dominating functions on \(G\) with the property that \(\sum_{i=1}^d f_i(v)\leq 2\) for each \(v\in V(G)\) is called a restrained Roman (Italian) dominating family (of functions) on \(G\). The maximum number of functions in a restrained Roman (Italian) dominating family on \(G\) is the restrained Roman (Italian) domatic number of \(G\), denoted by \(d_{rR}(G)\) \((d_{rI}(G))\). The author in this paper has presented a new sharp bound on \(d_{rR}(G)\) and \(d_{rI}(G)\). In addition, these parameters for some classes of graphs are determined.