Critical independent sets and Konig--Egervary graphs

is the subgraph of G spanned by X. By G -W we mean the subgraph G[V -W ] , if W ⊂ V (G). For F ⊂ E(G), by G -F we denote the partial subgraph of G obtained by deleting the edges of F , and we use Ge, if W = {e}. The neighborhood of a vertex v ∈ V is the set N (v) = {w : w ∈ V and vw ∈ E}, while N (A) = ∪{N (v) : v ∈ A} and

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 us denote the set {S : S is a maximum independent set of G} by Ω(G), and let core(G) = ∩{S : S ∈ Ω(G)} [17]. [16].

Theorem 1.1 [22] Every local maximum independent set of a graph is a subset of a maximum independent set.

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.

It is well-known that

We attribute this definition to Deming [6], and Sterboul [25]. These graphs were studied in [3,11,15,18,19,20,21,24], and generalized in [2,23].

According to a well-known result of König [10], and Egerváry [8], any bipartite graph is a König-Egerváry graph. This class includes non-bipartite graphs as well (see, for instance, the graphs H 1 and H 2 in Figure 1).

It is easy to see that if G is a König-Egerváry graph, then α(G) ≥ µ(G), and that a graph G having a perfect matching is a König-Egerváry graph if and only if α(G) = µ(G).

The number

, and the critical independence number α c (G) is the cardinality of a maximum critical independent set [26]. Clearly, α c (G) ≤ α(G) holds for any graph G. It is known that the problem of finding a critical independent set is polynomially solvable [1,26]. Proposition 1.2 [13] If S is a critical independent set, then there is a matching from N (S) into S.

If S is an independent set of a graph G and H = G -S, then we write G = S * H. Evidently, any graph admits such representations. For instance, if

Let M be a maximum matching of a graph G. To adopt Edmonds’s terminology [7], we recall the following terms for G relative to M . An alternating path from a vertex x to a vertex y is a x, y-path whose edges are alternating in and not in M . A vertex x is exposed relative to M if x is not the endpoint of a heavy edge. An odd cycle C with

The vertex x 0 is the base of the blossom. The stem is an even length alternating path joining the base of a blossom and an exposed vertex for M . The base is the only common vertex to the blossom and the stem. A flower is a blossom and its stem. A posy consists of two (not necessarily disjoint) blossoms joined by an odd length alternating path whose first and last edges belong to M . The endpoints of the path are exactly the bases of the two blossoms. The following result of Sterboul, characterizes König-Egerváry graphs in terms of forbidden configurations.

Theorem 1.4 [25] For a graph G, the following properties are equivalent:

(i) G is a König-Egerváry graph;

(ii) there exist no flower and no posy relative to some maximum matching M ; (iii) there exist no flower and no posy relative to any maximum matching M .

In [20] is given a characterization of König-Egerváry graphs having a perfect matching, in terms of certain forbidden subgraphs with respect to a specific perfect matching of the graph. In [12] is given the following characterization of König-Egerváry graphs in terms of excluded structures. Theorem 1.5 [12] Let M be a maximum matching in a graph G. Then G is a König-Egerváry graph if and only if G does not contain one of the forbidden configurations, depicted in Figure 2, with respect to M .

Figure 2: Forbidden configurations. The vertex v is not adjacent to the matching edges (namely, dashed edges).

In [14] it was shown that G is a König-Egerváry graph if and only if α c (G) = α(G), thus giving a positive answer to the Graffiti.pc 329 conjecture [5].

The deficiency of G, denoted by def (G), is defined as the number of exposed vertices relative to a maximum matching [21]. In other words, def

In this paper we prove that the critical difference for a König-Egerváry graph G is given by

and using this finding, we show that G is a König-Egerváry graph if and only if each of its maximum independent sets is critical.

Proposition 2.1 Every critical independent set is a local maximum independent set.

Proof. Suppose, on the contrary, that there is a critical independent set S such that S / ∈ Ψ(G), i.e., there exists some independent set A ⊆ N [S], larger than S. It follows that |A ∩ N (S)| > |S -S ∩ A|, and this contradicts the fact that, according to Proposition 1.2, there is a matching from A ∩ N (S) to S, in fact, from A ∩ N (S) to S -S ∩ A.

The converse of Proposition 2.1 is

…(Full text truncated)…