## Abstract

Let G= (V, E) be a graph with no isolated vertices. A vertex v totally dominates a vertex w (w≠ v), if v is adjacent to w. A set D⊆ V called a total dominating set of G if every vertex v∈ V is totally dominated by some vertex in D. The minimum cardinality of a total dominating set is the total domination number of G and is denoted by γ_{t}(G) . A total 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 χ_{td}(G) of G is the 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 Coloring problem for various graph classes, including trees, cographs and chain graphs. First, we characterize the trees having χ_{td}(T) = γ_{t}(T) + 1 , which completes the characterization of trees achieving all possible values of χ_{td}(T) . Also, we show that for a cograph G, χ_{td}(G) can be computed in linear-time. Moreover, we show that 2 ≤ χ_{td}(G) ≤ 4 for a chain graph G and then we characterize the class of chain graphs for every possible value of χ_{td}(G) in linear-time.

Original language | English |
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Article number | 128 |

Journal | Graphs and Combinatorics |

Volume | 39 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2023 |

## Keywords

- Bipartite graphs
- Chordal graphs
- Cographs
- Planar graphs
- Total dominator coloring

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics