Regular graphs with equal weakly connected domination number and matching number

Let G be a connected graph with vertex set V(G). A weakly connected dominating set W of G is a dominating set of G such that the subgraph consisting of V(G) and all edges incident with vertices in W is connected. The minimum cardinality among all weakly connected dominating sets of G is called the weakly connected domination number, denoted by γ_w(G). Bermudo, Dettlaff, and Lemańska (Discrete Appl. Math., 304 (2021), 153–163) posed a conjecture which says that if G is a regular graph of order |V(G)|≥3, then γ_w(G)=μ(G) if and only if G is a cycle, where μ(G) denotes the matching number of G. In this paper, we show that the conjecture is true.