Complexity of Total Dominator Coloring in Graphs

Abstract

Let G = (V,E) be a graph with no isolated vertices. A vertex v totally dominates a vertexw(w= v),ifvisadjacenttow.AsetD ⊆ V calledatotaldominatingsetofG if everyvertexv ∈ V istotallydominatedbysomevertexin D.Theminimumcardinality of a total dominating set is the total domination number of G and is denoted by γt(G). Atotal dominator coloring of graph G is a proper coloring of vertices of G, so that each vertex totally dominates some color class. The total dominator chromatic number χtd(G) of G isthe least number of colors required for a total dominator coloring of G. The Total Dominator Coloring problem is to find a total dominator coloring of G using the minimum number of colors. It is known that the decision version of this problem is NP-complete for general graphs. We show that it remains NP-complete even when restricted to bipartite, planar and split graphs. We further study the Total Dominator Coloringproblem for various graph classes, including trees, cographs and chain graphs. First, we characterize the trees having χtd(T) = γt(T) + 1, which completes the characterization of trees achieving all possible values of χtd(T).Also, we show that for a cograph G, χtd(G) can be computed in linear-time. Moreover, we show that 2 ≤ χtd(G) ≤ 4 for a chain graph G and then we characterize the class of chain graphs for every possible value of χtd(G) in linear-time