Minus \(k\)-subdomination in graphs (Q2713647)

A minus \(k\)-subdominating function on a graph \(G\) with the vertex set \(V(G)\) is a function \(f : V(G) \rightarrow \{-1,0,1\}\) such that there are at least \(k\) vertices \(v\) of \(G\) with the property that the sum \(f[v]\) of values of \(f\) over the closed neighbourhood of \(v\) is at least 1. The minimum of the sum \(f(V(G))\) of values of \(f\) over \(V(G)\), taken over all minus \(k\)-subdominating functions on \(G\), is the minus \(k\)-subdomination number \(\gamma ^{-1 \circ 1}_{ks}(G)\) of \(G\). Its properties are studied. In particular, the values \(\gamma ^{-1\circ 1}_{ks}(G)\) for paths \(P_n\) of length \(n\) are calculated and the least number of vertices of a connected graph \(G\) for which \(\gamma ^{-1\circ 1}_{ks} (G) = -m\) is determined.