Bipartite graphs with close domination and k-domination numbers

Abstract

Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq 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 {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 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.

Lict k-Domination in Graphs

“…The Concept of k-domination and its generalization as total k-domination number γ kt (G) used in (9) strong product graph to obtain several bounds for the γ kt (G) of the strong product of two graphs. As in 2020, Ekinci G.B., Bujtas C. characterized bipartite graphs satisfying the equality for k ≥ 3 and presented a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily if k = 3 (10) . In 2022 (11) , author studied relationships between the k-tuple domination number and parameters of graph and different dominations.…”

Some new results on the k-tuple domination number of graphs

“…Observe that the 1-tuple dominating set of G is the same as the dominating set of G. The k-tuple domination number of G, denoted by γ ×k (G), is the minimum cardinality among all k-tuple dominating sets of G. For a comprehensive survey on k-domination and k-tuple domination in graphs, we suggest the chapter [9] due to Hansberg and Volkmann. In addition, some recent results on these parameters can be found in [1,2,5,13,16].…”

On the equality of domination number and 2-domination number

“…In a follow-up paper of the present work [12], we studied the analogous problem for each k ≥ 3. There we gave a characterization for connected bipartite graphs satisfying γ k (G) = γ(G) + k − 2 and ∆(G) ≥ k. This result is based on the notion of the k-uniform "underlying hypergraph" that corresponds to the underlying graph, as defined here, if k = 2.…”