On the Intersection of All Critical Sets of a Unicyclic Graph

A set S is independent in a graph G if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. If alpha(G)+mu(G)=|V|, then G=(V,E) is called a Konig-Egervary graph. The number d_{c}(G)=max{|A|-|N(A)|} is called the critical difference of G (Zhang, 1990). By core(G) (corona(G)) we denote the intersection (union, respectively) of all maximum independent sets, while by ker(G) we mean the intersection of all critical independent sets. A connected graph having only one cycle is called unicyclic. It is known that ker(G) is a subset of core(G) for every graph G, while the equality is true for bipartite graphs (Levit and Mandrescu, 2011). For Konig-Egervary unicyclic graphs, the difference |core(G)|-|ker(G)| may equal any non-negative integer. In this paper we prove that if G is a non-Konig-Egervary unicyclic graph, then: (i) ker(G)= core(G) and (ii) |corona(G)|+|core(G)|=2alpha(G)+1. Pay attention that |corona(G)|+|core(G)|=2alpha(G) holds for every Konig-Egervary graph.

💡 Research Summary

The paper investigates the relationship between three fundamental vertex sets in graph theory—core, corona, and the kernel (ker)—within the class of unicyclic graphs, i.e., connected graphs that contain exactly one cycle. After recalling standard definitions (independent set, independence number α(G), matching, matching number µ(G), König‑Egerváry graphs, critical difference d_c(G), critical independent sets, core(G) as the intersection of all maximum independent sets, corona(G) as their union, and ker(G) as the intersection of all critical independent sets), the authors focus on how these sets interact when the underlying graph is unicyclic.

A key structural observation is that any unicyclic graph G can be decomposed into its unique cycle C and a collection of trees T_x attached to vertices of C. Each tree T_x is bipartite, which implies ker(T_x)=core(T_x) by known results for bipartite graphs. The paper builds on this decomposition to derive several new results.

First, Lemma 2.1 shows that for a non‑König‑Egerváry unicyclic graph, no vertex of the cycle belongs to core(G) and there exists a matching from N(core(G)) into core(G). This follows from the fact that every edge of C is α‑critical in the non‑König‑Egerváry case, forcing each cycle vertex to appear in some maximum independent set but never in all of them.

The authors then treat the two possible categories separately:

• For König‑Egerváry unicyclic graphs, Theorem 2.2 (a known result) states that N(core(G)) = V(G) \ corona(G) and that |corona(G)| + |core(G)| = 2α(G). This equality holds for all König‑Egerváry graphs, not only unicyclic ones.

• For non‑König‑Egerváry unicyclic graphs, Theorem 2.3 proves a precise description of corona(G): it consists of all cycle vertices together with the coronas of the attached trees, i.e.,
  corona(G) = V(C) ∪ ⋃_{x∈N₁(C)} corona(T_x).
  Moreover, every vertex of G belongs either to corona(G) or to N(core(G)), establishing the covering relation corona(G) ∪ N(core(G)) = V(G).

The central quantitative result is Theorem 2.4, which establishes the tight bound
2α(G) ≤ |corona(G)| + |core(G)| ≤ 2α(G) + 1,
and shows that the upper bound is attained exactly when G is a non‑König‑Egerváry unicyclic graph. The proof hinges on the fact that in this case α(G) + µ(G) = |V(G)| – 1, i.e., the sum falls short of the vertex count by one, and on a careful counting argument involving matchings from N(core(G)) into core(G) and from S \ core(G) into corona(G) \ S for a maximum independent set S.

The most striking structural theorem is Theorem 2.5, which proves that for any non‑König‑Egerváry unicyclic graph, the kernel coincides with the core: ker(G) = core(G). The argument proceeds by contradiction: assuming some tree T_q contributes a vertex to ker(G) that is not in core(T_q) would lower the critical difference of ker(G), contradicting its maximality. Since each tree satisfies ker(T_x)=core(T_x) and the cycle contributes no vertices to ker(G) (by Lemma 2.1(i)), the equality follows.

Finally, the paper discusses the variability of |core(G)| – |ker(G)| in non‑bipartite König‑Egerváry unicyclic graphs, showing that this difference can be any non‑negative integer (Figure 5). It poses two open problems: characterizing non‑bipartite König‑Egerváry unicyclic graphs with core(G)=ker(G), and characterizing all graphs for which the inequality 2α(G) ≤ |corona(G)| + |core(G)| ≤ 2α(G)+1 holds with equality at either bound.

In summary, the authors extend the known relationships between core, corona, and ker from bipartite and König‑Egerváry graphs to the broader class of unicyclic graphs. They demonstrate that the equality ker(G)=core(G) holds precisely for non‑König‑Egerváry unicyclic graphs, and that the sum of the sizes of corona and core jumps from 2α(G) to 2α(G)+1 exactly when the graph fails to be König‑Egerváry. These findings deepen the understanding of how independent sets and matchings interact in graphs with a single cycle and provide a foundation for further investigations into critical structures of more complex graph families.