Fair total domination number in cactus graphs

Abstract

For k ≥ 1, a k-fair total dominating set (or just kFTD-set) in a graph G is a total dominating set S such that |N (v) ∩ S| = k for every vertex v ∈ V \S. The k-fair total domination number of G, denoted by f td k (G), is the minimum cardinality of a kFTD-set. A fair total dominating set, abbreviated FTD-set, is a kFTD-set for some integer k ≥ 1. The fair total domination number of a nonempty graph G, denoted by f td(G), of G is the minimum cardinality of an FTD-set in G. In this paper, we present upper bounds for the 1-fair total domination number of cactus graphs, and characterize cactus graphs achieving equality for the upper bounds.