A characterization of Konig-Egervary graphs using a common property of all maximum matchings

If X ⊂ V , then G[X] 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}.

If A, B ⊂ V and A ∩ B = ∅, then (A, B) stands for the set

The neighborhood of a vertex v ∈ V is the set

and N (A) = ∪{N (v) : v ∈ A}, while N [A] = A ∪ N (A) for A ⊂ V . By P n , C n , K n we mean the chordless path on n ≥ 3, the chordless cycle on n ≥ 4 vertices, and respectively the complete graph on n ≥ 1 vertices.

A set S of vertices is independent if no two vertices from S are adjacent. An independent set of maximum size will be referred to as a maximum independent set of G. The independence number of G, denoted by α(G), is the cardinality of a maximum independent set of G.

By Ind(G) we mean the set of all independent sets of G. Let Ω(G) denote the set of all maximum independent sets of G [15], and core(G) = ∩{S : S ∈ Ω(G)}.

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

Egerváry graph (a K-E graph, for short). We attribute this definition to Deming [5], and Sterboul [25]. These graphs were studied in [3,11,21,22,24], and generalized in [2,23]. Several properties of K-E graphs are presented in [14,16,17,18,19].

] is a K-E graph with a perfect matching and each maximum matching of H can be enlarged to a maximum matching of G.

According to a well-known result of König [10] and Egerváry [7], every bipartite graph is a K-E graph. This class includes also some non-bipartite graphs (see, for instance, the graph from Figure 1).

, and that a graph G having a perfect matching is a K-E graph if and only if α(G) = µ(G).

If S is an independent set of a graph G and H = G[V -S], then we write G = S * H. Clearly, any graph admits such representations. However, some particular cases are of special interest. For instance, if Let M be a maximum matching of a graph G. To adopt Edmonds’s terminology, [6], we recall the following terms for G relative to M . The edges in M are heavy, while those not in M are light. An alternating path from a vertex x to a vertex y is a x, y-path whose edges are alternating light and heavy. A vertex x is exposed relative to M if x is not the endpoint of a heavy edge. An odd cycle C with V (C) = {x 0 , x 1 , …, x 2k } and

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.

Theorem 1.3 [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 every maximum matching M .

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

In [9], Gavril defined the so-called red/blue-split graphs, as a common generalization of K-E and split graphs. Namely, G is a red/blue-split graph if its edges can be colored in red and blue such that V (G) can be partitioned into a red and a blue independent set (where red or blue independent set is an independent set in the graph made of red or blue edges). In [12], Korach et al. described red/blue-split graphs in terms of excluded configurations, which led them to the following characterization of K-E graphs.

Theorem 1.4 [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 .

In [21], Lovasz gives a characterization of K-E graphs having a perfect matching, in terms of certain forbidden subgraphs with respect to a specific perfect matching of the graph.

The problem of recognizing K-E graphs is polynomial as proved by Deming [5], of complexity O(|V (G)| |E(G)|). Gavril [9] has described a recognition algorithm for K-E graphs of complexity O(|V (G)| + |E(G)|). The problem of finding a maximum independent set in a K-E graph is polynomial as proved by Deming [5].

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].

In [13] it was shown that G