Critical Sets in Bipartite Graphs

Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent, alpha(G) is the size of a maximum independent set, and core(G) is the intersection of all maximum independent sets. The number d(X)=|X|-|N(X)| is the difference of the set X, and d_{c}(G)=max{d(I):I is an independent set} is called the critical difference of G. A set X is critical if d(X)=d_{c}(G). For a graph G we define ker(G) as the intersection of all critical independent sets, while diadem(G) is the union of all critical independent sets. For a bipartite graph G=(A,B,E), with bipartition {A,B}, Ore defined delta(X)=d(X) for every subset X of A, while delta_0(A)=max{delta(X):X is a subset of A}. Similarly is defined delta_0(B). In this paper we prove that for every bipartite graph G=(A,B,E) the following assertions hold: d_{c}(G)=delta_0(A)+delta_0(B); ker(G)=core(G); |ker(G)|+|diadem(G)|=2*alpha(G).

💡 Research Summary

The paper investigates the interplay between several classical and newly introduced parameters of bipartite graphs, focusing on independent sets, matchings, and the notions of “critical” and “core” structures. After recalling standard definitions—independent set, maximum independent set size α(G), core(G) as the intersection of all maximum independent sets, and the difference function d(X)=|X|−|N(X)|—the authors define the critical difference d_c(G)=max_X d(X) and identify critical sets as those attaining this maximum. They further introduce ker(G) as the intersection of all critical independent sets and diadem(G) as their union.

For a bipartite graph G=(A,B,E) the paper adopts Ore’s notation δ(X)=d(X) for subsets X of a single part, and defines δ₀(A)=max_{X⊆A} δ(X) and δ₀(B)=max_{Y⊆B} δ(Y). The main results are threefold:

1. Critical Difference Decomposition – The authors prove that d_c(G)=δ₀(A)+δ₀(B). Using the König‑Egerváry property (α(G)+μ(G)=|V| for bipartite graphs), they derive equivalent expressions α(G)=|A|+δ₀(B)=|B|+δ₀(A)=μ(G)+δ₀(A)+δ₀(B)=μ(G)+d_c(G). This links the global critical difference to the maximal “deficiency” of each bipartition.

2. Equality of Ker and Core – The central theorem shows ker(G)=core(G) for every bipartite graph. The proof hinges on three equivalent characterizations of ker(G): (i) minimality among d‑critical sets, (ii) the non‑existence of a non‑empty Y⊆N(ker) with |N(Y)∩ker|=|Y|, and (iii) the existence of a matching from N(ker) into ker−v for each v∈ker. By assuming a non‑empty core, employing a maximum matching M that matches V−S into a maximum independent set S, and constructing a minimal violating set Z⊆N(core), the authors derive a contradiction through a series of claims about connectivity, independence, and extendability. Consequently, any vertex belonging to all maximum independent sets must also belong to every critical independent set, establishing ker=core.

3. Ker–Diadem Size Relation – Combining the previous results with the expression for α(G), the authors obtain |ker(G)|+|diadem(G)|=2α(G). Since diadem(G) is the union of all critical independent sets, this identity reveals that the sizes of the intersection and the union of critical independent sets together account for twice the size of a maximum independent set.

Supporting lemmas elucidate the structure of A‑critical and B‑critical subsets. Lemma 2.3 proves that for an A‑critical X and a B‑critical Y, the cardinalities |X∩N(Y)| and |N(X)∩Y| coincide and admit a perfect matching between these two parts. Corollary 2.4 shows that ker_A and ker_B are disjoint from each other’s neighborhoods, reinforcing the separation of critical structures across the bipartition.

The paper also revisits known facts: Theorem 1.1 establishes that d_c(G)=id_c(G) (the critical difference equals the maximum difference over independent sets), and that the function d is supermodular. Theorem 1.2 recalls that in König‑Egerváry graphs a maximum matching matches the complement of any maximum independent set into the set itself, and that such a maximum independent set is d‑critical.

Overall, the work provides a unified framework that connects matching theory, supermodular set functions, and independent set extremality in bipartite graphs. By proving that the kernel of critical independent sets coincides with the classical core, and by quantifying the relationship between kernel, diadem, and α(G), the authors deepen our understanding of the internal geometry of bipartite graphs and open avenues for algorithmic exploitation of these structural properties.