A characterization of graphs attaining a lower bound on the dissociation number

The dissociation number diss(G) of a graph G is defined as the maximum cardinality of a vertex subset D⊆ V(G) such that the induced subgraph $G[D]$ has maximum degree at most $1$. In 2023, Bock, Pardey, Penso and Rautenbach [J. Graph Theory 103 (2023), 661--673] showed that diss(G)≥ n-1/3(m+k+c₁) where n, m, k, c₁ denote the numbers of vertices, edges, components and induced cycles of length $1$ modulo $3$ in G and they also established a complete characterization of extremal graphs under the vertex-disjoint cycle constraint. Although this partial characterization offers very valuable structural insights, it excludes many simple extremal graphs. So they posed a natural question: How can all extremal graphs be comprehensively characterized without restrictions? In this paper, we give a recursive characterization of all graphs attaining equality in this bound.