Signed total double Roman dominating functions in graphs
L. Shahbazi, Hossein Abdollahzadeh Ahangar, Seyed Mahmoud Sheikholeslami · AKCE International Journal of Graphs and Combinatorics · 2024
A signed total double Roman dominating function (STDRDF) on an isolated-free graph G=(V,E) is a function f:V(G)→{−1,1,2,3} such that (i) every vertex v with f(v)=−1 has at least two neighbors assigned 2 under f or one neighbor w with f(w) = 3, (ii) every vertex v with f(v) = 1 has at least one neighbor w with f(w)≥2 and (iii) ∑u∈N(v)f(u)≥1 holds for any vertex v. The weight of a STDRDF is the value f(V(G))=∑u∈V(G)f(u). The signed total double Roman domination number γsdRt(G) is the minimum weight of a STDRDF on G. In this article, we provide various bounds on γsdRt(G) and we show that the corresponding decision problem is NP-complete for bipartite and chordal graphs. In addition, we determine the signed total double Roman domination number of some classes of graphs including cycles, complete graphs and complete bipartite graphs.