2020/05/16 by Gülnaz Boruzanlı Ek̇inċi, Ekinci, Gülnaz Boruzanlı, Csilla Bujtás +1
Computer Science · Mathematics · #05C69 #05C75 #68Q25 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2005.07835
openalex publication_date 2020/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a positive integer and let G be a graph with vertex set V(G). A subset D ⊆ V(G) is a k-dominating set if every vertex outside D is adjacent to at least k vertices in D. The k-domination number γk(G) is the minimum cardinality of a k-dominating set in G. For any graph G, we know that γk(G) ≥ γ(G)+k-2 where Δ(G)≥ k≥ 2 and this bound is sharp for every k≥ 2. In this paper, we characterize bipartite graphs satisfying the equality for k≥ 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3. We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.