Regular graphs with equal matching number and independence number

Let r >= 3 be an integer and G be a graph. Let delta(G), Delta(G), alpha(G), and mu(G) denote the minimum degree, maximum degree, independence number, and matching number of G, respectively. Recently, Caro, Davila and Pepper proved delta(G)alpha(G) <= Delta(G)mu(G). Mohr and Rautenbach characterized the extremal graphs within the classes of all non-regular graphs and 3-regular graphs. In this note, we characterize the extremal graphs within the classes of all r-regular graphs in terms of the Gallai-Edmonds Structure Theorem, which extends Mohr and Rautenbach's result. (C) 2021 Elsevier B.V. All rights reserved.