Critical independent sets and Konig--Egervary graphs

Let alpha(G) be the cardinality of a independence set of maximum size in the graph G, while mu(G) is the size of a maximum matching. G is a Konig–Egervary graph if its order equals alpha(G) + mu(G). The set core(G) is the intersection of all maximum independent sets of G (Levit & Mandrescu, 2002). The number def(G)=|V(G)|-2*mu(G) is the deficiency of G (Lovasz & Plummer, 1986). The number d(G)=max{|S|-|N(S)|:S in Ind(G)} is the critical difference of G. An independent set A is critical if |A|-|N(A)|=d(G), where N(S) is the neighborhood of S (Zhang, 1990). In 2009, Larson showed that G is Konig–Egervary graph if and only if there exists a maximum independent set that is critical as well. In this paper we prove that: (i) d(G)=|core(G)|-|N(core(G))|=alpha(G)-mu(G)=def(G) for every Konig–Egervary graph G; (ii) G is Konig–Egervary graph if and only if every maximum independent set of G is critical.

💡 Research Summary

The paper investigates the relationship between critical independent sets, the core of a graph, and König‑Egervary (KE) graphs. After recalling standard definitions—α(G) as the size of a maximum independent set, μ(G) as the size of a maximum matching, core(G) as the intersection of all maximum independent sets, the deficiency def(G)=|V(G)|−2μ(G), and the critical difference d(G)=max_{S∈Ind(G)}(|S|−|N(S)|)—the authors focus on the class of KE graphs, which satisfy |V(G)|=α(G)+μ(G).

The first main result (Theorem 2.4) shows that for any KE graph G the four quantities coincide: d(G)=|core(G)|−|N(core(G))|=α(G)−μ(G)=def(G).
The proof proceeds by first establishing the universal inequality |S|−|N(S)|≤α(G)−μ(G) for any independent set S, using the maximality of a matching M. Then, because core(G) is itself independent, the authors obtain |core(G)|−|N(core(G))|≤α(G)−μ(G). The KE condition allows the decomposition |V|=|core|+|N(core)|+|V−N