Signed total domination in graphs.
The paper continues the study of the signed total domination in graphs which was started by the reviewer. Let \(G\) be a graph with vertex set \(V\). For each \(v\in V\) let \(N(v)\) be the open neighbourhood of \(v\) in \(G\), i.e. the set of all vertices adjacent to \(v\) in \(G\). If \(f\) is a mapping of \(V\) into a number set and \(S\subseteq V\), then \(f(S)= \sum_{X\in V} f(x)\). The sum \(w(f)= f(V)\) is the weight of \(f\). A function \(f: V\to \{-1,1\}\) is called a signed total dominating function on \(G\) (shortly STDF), if \(f(N(v))\geq 1\) for each \(v\in G\). It is minimal, if no \(g< f\) is a STDF. The minimum weight of an STDF on \(G\) is the signed total domination number \(\gamma^s_t(G)\) of \(G\). The maximum weight of a minimal STDF on \(G\) is the upper signed total domination number \(\Gamma^s_t(G)\) of \(G\). The paper presents bounds for these numbers in terms of the minimum degree \(\delta\) and the maximum degree \(\Delta\). Special attention is paid to regular graphs, trees and bipartite graphs. The problem of recognizing whether \(\gamma^s_t(H)\leq j\) for a given graph \(H\) and positive integer \(j\) is proved to be NP-complete.
scientific article; zbMATH DE number 2055735