Total dominating sequences in graphs

Boštjan Brešar, Michael A. Henning, Douglas F. Rall

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)

Abstract

A vertex in a graph totally dominates another vertex if they are adjacent. A sequence of vertices in a graph G is called a total dominating sequence if every vertex v in the sequence totally dominates at least one vertex that was not totally dominated by any vertex that precedes v in the sequence, and at the end all vertices of G are totally dominated. While the length of a shortest such sequence is the total domination number of G, in this paper we investigate total dominating sequences of maximum length, which we call the Grundy total domination number, γgrt(G), of G. We provide a characterization of the graphs G for which γgrt(G)=|V(G)| and of those for which γgrt(G)=2. We show that if T is a nontrivial tree of order n with no vertex with two or more leaf-neighbors, then γgrt(T)≥23(n+1), and characterize the extremal trees. We also prove that for k≥3, if G is a connected k-regular graph of order n different from Kk,k, then γgrt(G)≥(n+Fk2F-2)/(k-1) if G is not bipartite and γgrt(G)≥(n+2Fk2F-4)/(k-1) if G is bipartite. The Grundy total domination number is proven to be bounded from above by two times the Grundy domination number, while the former invariant can be arbitrarily smaller than the latter. Finally, a natural connection with edge covering sequences in hypergraphs is established, which in particular yields the NP-completeness of the decision version of the Grundy total domination number.

Original languageEnglish
Pages (from-to)1665-1676
Number of pages12
JournalDiscrete Mathematics
Volume339
Issue number6
DOIs
Publication statusPublished - 6 Jun 2016

Keywords

  • Edge cover
  • Grundy total domination number
  • Total domination

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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