On the Core of a Unicyclic 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 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 {e = ab : a ∈ A, b ∈ B, e ∈ E}. The neighborhood of a vertex v ∈ V is the set N (v) = {w : w ∈ V and vw ∈ E}, and N (A) = ∪{N (v) : v ∈ A}, N [A] = A ∪ N (A) for A ⊂ V . By C n , K n we mean 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, 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 size of a maximum independent set of G. Let Ω(G) denote the family {S : S is a maximum independent set of G}, while core(G) = ∩{S : S ∈ Ω(G)} [11].

An edge e ∈ E(G) is α-critical whenever α(Ge) > α(G). Notice that the inequalities α(G) ≤ α(Ge) ≤ α(G) + 1 hold for each edge e.

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).

Theorem 1.1 [13] For every graph G no α-critical edge has an endpoint in N [core(G)].

It is well-known that [4], [19]. Several properties of König-Egerváry graphs are presented in [6], [9], [10], [12], [14], [16].

It is known that every bipartite graph is a König-Egerváry graph as well [5], [8]. This class includes also non-bipartite graphs (see, for instance, the graph G in Figure 1). Theorem 1.2 If G is a König-Egerváry graph, then (i) [12] every maximum matching matches N (core(G)) into core(G);

(ii) [13]

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

The graph G is called unicyclic if it is connected and has a unique cycle, which we denote by C = (V (C), E (C)). Let Unicyclic graphs keep enjoying plenty of interest, as one can see, for instance, in [1], [3], [7], [15], [18], [20], [21].

In this paper we analyze the structure of core(G) for a unicyclic graph G.

If G is a unicyclic graph, then there is an edge e ∈ E (C), such that µ(Ge) = µ(G), because for each pair of edges, consecutive on C, at most one could be µ-critical. Let us mention that α(G) ≤ α(Ge) ≤ α(G) + 1 holds for each edge e ∈ E (G). Every edge of the unique cycle could be α-critical; e.g., the graph G from Figure 2, which has also additional α-critical edges (e.g., the edge uv).

Let us notice that the bipartite graph T x from Figure 2 has only two maximum matchings, namely, M 1 = {ax, uv} and M 1 = {bx, uv}, while each vertex of core(T x ) = {a, b} is not saturated by one of these matchings.

Lemma 2.1 For every bipartite graph G, a vertex v ∈ core(G) if and only if there exists a maximum matching that does not saturate v.

In other words, there is a maximum matching in G not saturating v.

Conversely, suppose that there exists a maximum matching in G that does not saturate v. Since, by Theorem 1.

While C 2k , k ≥ 2, has no α-critical edge at all, each edge of every odd cycle C 2k-1 , k ≥ 2, is α-critical. This property is partially inherited by unicyclic graphs. Lemma 2.5 Let G be a unicyclic graph of order n. Then n -1 = α(G) + µ(G) if and only if each edge of its unique cycle is α-critical.

Proof. Assume that n -1 = α(G) + µ(G). Since G is connected, for each e ∈ E(C) the graph Ge is a tree. Hence, we have

In other words, every e ∈ E(C) is α-critical.

Conversely, let e ∈ E (C) be such that µ(Ge) = µ(G); such an edge exists, because no two consecutive edges on C could be µ-critical. Since e is α-critical, and Ge is a tree, we infer that

and this completes the proof.

Combining Lemma 2.5 and Theorem 1.1, we infer the following.

Remark 2.7 Corollary 2.6 is true also for some unicyclic König-Egerváry graphs; e.g., the graph H 1 from Figure 3. However, the König-Egerváry graph H 2 from the same

Lemma 2.8 Let G be a unicyclic graph of order n. If there exists some x ∈ N 1 (C), such that x ∈ core(T x ), then G is a König-Egerváry graph.

Proof. Let x ∈ core(T x ), y ∈ N (x) ∩ V (C), and z ∈ N (y) ∩ V (C). Suppose, to the contrary, that G is not a König-Egerváry graph. By Lemmas 2.3 and 2.5, the edge yz is α-critical. Hence y / ∈ core(G), which implies that α(G) = α(Gy). In accordance with Lemma 2.1, there exists a maximum matching M x of T x not saturating x. Combining M x with a maximum matching of Gy -T x we get a maximum matching M y of Gy. Hence M y ∪ {xy} is a matching of G, which results in µ (G) ≥ µ (Gy) + 1. Therefore, using Lemma 2.3 and having in mind that Gy is a forest of order n -1, we get the following contrad

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