2023/03/03 by Michael A. Henning, Henning, Michael A., Kusum +4
Computer Science · Decision Sciences · Neuroscience · #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Complexity (cs.CC) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Nuclear Receptors and Signaling #Scheduling and Timetabling Solutions
paper · pdf · doi:10.48550/arxiv.2303.01746
openalex publication_date 2023/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G=(V,E) be a graph with no isolated vertices. A vertex v totally dominate 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 γt(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 χtd(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 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 give characterization of chain graphs for every possible value of χtd(G) in linear-time.