On the Structure of the Minimum Critical Independent Set of a Graph

The neighborhood of a vertex v ∈ V is the set N (v) = {w : w ∈ V and vw ∈ E}, while the closed neighborhood of v ∈ V is N [v] = N (v)∪{v}; in order to avoid ambiguity, we use also N G (v) instead of N (v). The neighborhood of A ⊆ V is denoted by N (A) = N G (A) = {v ∈ V : N (v) ∩ A = ∅}, and N [A] = N (A) ∪ A.

A set S ⊆ V (G) is independent if no two vertices from S are adjacent, and by Ind(G) we mean the set of all the independent sets of G.

An independent set of maximum size will be referred to as a maximum independent set of G, and the independence number of G is α(G) = max{|S| : S ∈ Ind(G)}. Let Ω(G) denote the family of all maximum independent sets, and core(G) = ∩{S : S ∈ Ω(G)} [4].

A matching is a set of non-incident edges of G; a matching of maximum cardinality is a maximum matching, and its size is denoted by µ(G).

The number [7]. The number id c (G) = max{d(I) :

For a graph G, let denote ker(G) = ∩ {S : S is a critical independent set }. It is known that ker(G) ⊆ core(G) is true for every graph [5], while the equality holds for bipartite graphs [6].

For instance, the graph G from Figure 1 has

In addition, ker(G) = {v 1 , v 2 }, and core(G) is a critical set.

It is easy to see that all pendant vertices are included in every maximum critical independent set. It is known that the problem of finding a critical independent set is polynomially solvable [1,7]. Theorem 1.2 For a graph G = (V, E), the following assertions are true:

(i) [5] the function d is supermodular, i.e.,

(ii) [5] G has a unique minimal critical independent set, namely, ker(G). (iii) [3] there is a matching from N (S) into S, for every critical independent set S.

In this paper we characterize ker(G). In addition, a number of properties of ker(G) are presented as well.

Deleting a vertex from a graph may decrease, leave unchanged or increase its critical difference. For instance,

, where G is depicted in Figure 1.

Proposition 2.1 Let G = (V, E) and v ∈ V . Then the following assertions hold:

Consequently, we infer that

) and this contradicts the minimality of ker(G). Therefore, N (ker(G) -{v}) = N (ker(G)) and hence

On the other hand, there is no matching from N (S) into Sv 3 . The case of the critical independence set ker(G) is more specific. Theorem 2.3 Let A be a critical independent set in a graph G. Then the following statements are equivalent:

Proof. (i) =⇒ (ii) By Theorem 1.2(iii), there is a matching, say M , from N (ker(G)) into ker(G). Suppose, to the contrary, that there is some non-empty set

It contradicts the fact that, by Theorem 1.2(ii), ker(G) is a minimal critical independent set, because

, there is a matching, say M , from N (A) into A. Since there are no edges connecting vertices belonging to ker(G) with vertices from N (A) -N (ker(G)), we obtain that M (N (A) -N (ker(G))) ⊆ Aker(G). Moreover, we have that

It means that the set N (A) -N (ker(G)) contradicts the hypothesis of (ii), because

Consequently, the assertion is true.

(ii) =⇒ (iii) By Theorem 1.2(iii), there is a matching, say M , from N (A) into A. Suppose, to the contrary, that there is no matching from N (A) into Av. Hence, by Hall’s Theorem, it implies the existence of a set B ⊆ N (A) such that |N (B) ∩ A| = |B|, which contradicts the hypothesis of (ii).

(iii) =⇒ (ii) Assume, to the contrary, that there is a non-empty subset B of N (A) such that |N (B) ∩ A| = |B|. Let v ∈ N (B) ∩ A. Hence, we obtain that

Then, by Hall’s Theorem, it is impossible to find a matching from N (A) into Av, in contradiction with the hypothesis of (iii).

Since ker(G) is a critical set, Theorem 1.2(iii) assures that there is a matching from N (ker(G)) into ker(G). The following result shows that there are at least two such matchings.

Corollary 2.4 For a graph G the following are true:

(i) every edge e ∈ (ker(G), N (ker(G))) belongs to a matching from N (ker(G)) into ker(G);

(ii) every edge e ∈ (ker(G), N (ker(G))) is not included in one matching from N (ker(G)) into ker(G) at least.

Proof. Let e = xy ∈ (ker(G), N (ker(G))), such that x ∈ ker(G). By Theorem 2.3(iii) there is a matching M from N (ker(G)) into ker(G)x, that matches y with some z ∈ ker(G)x. Clearly, M is a matching from N (ker(G)) into ker(G) that does not contain the edge e = xy, while (M -{yz}) ∪ {xy} is a matching from N (ker(G)) into ker(G), which includes the edge e = xy. Let us notice that the graphs G 1 , G 2 from Figure 2 have: ker(G 1 ) = core(G 1 ), ker(G 2 ) = {x, y, z} ⊂ core(G 2 ), and both core(G 1 ) and core(G 2 ) are critical sets of maximum size. The graph G 3 from Figure 2 has ker(G 3 ) = {u,

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