On graphs all of whose total dominating sequences have the same length
Discrete Mathematics, cilt.344, sa.9, 2021 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 344 Sayı: 9
- Basım Tarihi: 2021
- Doi Numarası: 10.1016/j.disc.2021.112492
- Dergi Adı: Discrete Mathematics
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, MathSciNet, zbMATH
- Anahtar Kelimeler: Grundy total domination number, Total domination number
- Ankara Yıldırım Beyazıt Üniversitesi Adresli: Evet
Özet
© 2021 Elsevier B.V.A sequence of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex v in the sequence has a neighbor which is adjacent to no vertex preceding v in the sequence, and at the end every vertex of G has at least one neighbor in the sequence. Minimum and maximum lengths of a total dominating sequence is the total domination number of G (denoted by γt(G)) and the Grundy total domination number of G (denoted by γgrt(G)), respectively. In this paper, we study graphs where all total dominating sequences have the same length. For every positive integer k, we call G a total k-uniform graph if every total dominating sequence of G is of length k, that is, γt(G)=γgrt(G)=k. We prove that there is no total k-uniform graph when k is odd. In addition, we present a total 4-uniform graph which stands as a counterexample for a conjecture by [11] and provide a connected total 8-uniform graph. Moreover, we prove that every total k-uniform, connected and false twin-free graph is regular for every even k. We also show that there is no total k-uniform chordal connected graph with k≥4 and characterize all total k-uniform chordal graphs.