A generalised upper bound for the \(k\)-tuple domination number (Q2470457)

Consider finite and undirected graphs without loops and multiple edges. Let \(V(G)\) be the set of vertices of \(G\), and \(N(x)\) be the neighbourhood of a vertex \(x\). A subset \(X\) of \(V(G)\) is called a dominating set if every vertex not in \(X\) is adjacent to a vertex in \(X\). The domination number is the minimum cardinality of a dominating set of \(G\). A set \(X\) is called a \(k\)-tuple dominating set of \(G\) if for every vertex \(x\) in \(V(G)\), the cardinality of \(N(v) {\cap} X\) is at least \(k\). The \(k\)-tuple dominating number is the minimum cardinality of a \(k\)-tuple dominating set of \(G\). In this paper, the authors provide an upper bound for the \(k\)-tuple dominating number that generalizes known upper bounds for the 2-tuple and 3-tuple dominating numbers.