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)|=2*alpha(G)+1. Pay attention that |corona(G)|+|core(G)|=2*alpha(G) holds for every Konig-Egervary graph.
Throughout this paper G = (V, E) is a simple (i.e., a finite, undirected, loopless and without multiple edges) graph with vertex set V = V (G) and edge set E = E(G). If X ⊂ V , then G[X] is the subgraph of G spanned by X. By G-W we mean the subgraph A set S of vertices is independent if no two vertices from S are adjacent, and an independent set of maximum size will be referred to as a maximum independent set. The independence number of G, denoted by α(G), is the cardinality of a maximum independent set of G.
Let core(G) = ∩{S : S ∈ Ω(G)} [9], and corona(G) = ∪{S : S ∈ Ω(G)} [2], where Ω(G) = {S : S is a maximum independent set of G}.
Theorem 1.1 [2] For every S ∈ Ω (G), there is a matching from S-core(G) into corona(G) -S. [21]. The number id c (G) = max{d(I) :
For a graph G, let denote ker(G) = ∩ {S : S is a critical independent set }. Theorem 1.3 If G is a graph, then (i) [13] ker (G) is a critical independent set and ker (G) ⊆ core(G); (ii) [14] ker (G) = core(G), whenever G is bipartite.
A matching (i.e., a set of non-incident edges of G) of maximum cardinality µ(G) is a maximum matching, and a perfect matching is one covering all vertices of G. An edge e ∈ E(G) is µ-critical provided µ(Ge) < µ(G).
It is well-known that ⌊n/2⌋ + 1 ≤ α(G) + µ(G) ≤ n hold for any graph G with n vertices. If α(G) + µ(G) = n, then G is called a König-Egerváry graph [3], [18]. Several properties of König-Egerváry graphs are presented in [8], [10], [12].
According to a celebrated result of König, [7], and Egerváry, [5], any bipartite graph is a König-Egerváry graph. This class includes also non-bipartite graphs (see, for instance, the graph G in Figure 1). The graph G is called unicyclic if it is connected and has a unique cycle, which we denote by
and
The following result shows that a unicyclic graph is either a König-Egerváry graph or each edge of its cycle is α-critical.
Theorem 1.6 [16] Let G be a unicyclic non-König-Egerváry graph. Then the following assertions are true:
Unicyclic graphs keep enjoying plenty of interest, as one can see , for instance, in [1], [4], [6], [11], [17], [19], [20].
In this paper we analyze the relationship between several parameters of a unicyclic graph G, namely, core(G), corona(G), ker (G).
(ii) there exists a matching from N (core(G)) into core(G).
Proof. (i) Let ab ∈ E (C). By Lemma 1.5(ii), the edge ab is α-critical. Hence there are S a , S b ∈ Ω (G), such that a ∈ S a and b ∈ S b . Since a / ∈ S b , it follows that a / ∈ core(G), and because a ∈ S a , we infer that N (a) ∩ core(G) = ∅. Consequently, we obtain that core(G) ∩ N [V (C)] = ∅.
(ii) If core(G) = ∅, then the conclusion is clear.
Assume that core(G) = ∅. By Theorem 1.4, in each tree T x there is a matching M x from N (core(T x )) into core(T x ). By part (i), we have that V (C) ∩ N [core(G)] = ∅. Taking into account Theorem 1.6(ii), we see that the union of all these matchings M x gives a matching from N (core(G)) into core(G).
It is worth mentioning that the assertion in Lemma 2.1(ii) is true for every König-Egerváry graph, by Theorem 1.4. The graph G 2 from Figure 3 shows that Lemma 2.1(i) may fail for unicyclic König-Egerváry graphs.
, it follows that there is some y ∈ N 1 (C), such that a ∈ V (T y ). According to Theorem 2.2(i), we know that V (T y ) = corona(T y ) ∪ N (core(T y )).
By Theorem 1.6(i), corona (T x ) ⊆ corona (G) for every x ∈ N 1 (C). Therefore, either a ∈ corona (T y ) ⊆ corona (G), or a ∈ N (core (T y )) ⊆ N (core (G)), because core (T y ) ⊆ core(G) in accordance with Theorem 1.6(iii).
Consequently, a ∈ corona (G) ∪ N (core (G)). In other words, we get that
As for the second equality, let us notice that V (C) ⊆ corona (G), by Case 1. If a ∈ corona (G) -V (C), then by Theorem 1.6(ii), there are S ∈ Ω (G) and b ∈ N 1 (C), such that a ∈ S ∩ V (T b ) ∈ Ω (T b ). Hence a ∈ corona (T x ), and therefore,
The graph G 2 from Figure 4 shows that the equality |corona(G)|+ |core(G)| = 2α (G) is not true for unicyclic non-König-Egerváry graphs.
Moreover, G is a non-König-Egerváry graph if and only if
Assume now that G is not a König-Egerváry graph. Let S ∈ Ω (G). According to Theorem 2.3 and Lemma 1.5(i), we infer that
Lemma 1.5 implies that there is a matching M from N (core (G)) into core (G), that can be enlarged to a maximum matching, say M 2 , of G.
Since M 2 matches µ (G) vertices from A = (corona(G) -S)∪(N (core (G))) by means of µ (G) edges, and |A| = µ (G)+1, it follows that M 2 -M matches |(corona(G) -S)|-1 vertices from A into Score (G), because M saturates N (core (G)) and no edge joins a vertex of core (G) to some vertex from corona(G) -S. Hence, taking into account that M ∪ M 1 is a matching of G, while M 2 is a maximum matching, we obtain
Finally, we infer that
and this completes the proof. Proof. Since T x is bipartite, by Theorem 1.3(ii) implies that ker (T x ) = core(T x ), for every x ∈ N 1 (C).
According to Theorems 1.3(i) and 1.6(iii), it follows that
Assume that A q = k
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