On some conjectures concerning critical independent sets of a graph

Let $G$ be a simple graph with vertex set $V(G)$. A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. For $X\subseteq V(G)$, the difference of $X$ is $d(X) = |X|-|N(X)|$ and an independent set $A$ is critical if $d(A) = \max \{d(X): X\subseteq V(G) \text{ is an independent set}\}$ (possibly $A=\emptyset$). Let $\text{nucleus}(G)$ and $\text{diadem}(G)$ be the intersection and union, respectively, of all maximum size critical independent sets in $G$. In this paper, we will give two new characterizations of K\"{o}nig-Egerv\'{a}ry graphs involving $\text{nucleus}(G)$ and $\text{diadem}(G)$. We also prove a related lower bound for the independence number of a graph. This work answers several conjectures posed by Jarden, Levit, and Mandrescu.


Introduction
In this paper G is a simple graph with vertex set V (G), |V (G)| = n, and edge set E(G). The set of neighbors of a vertex v is N G (v) or simply N (v) if there is no possibility of ambiguity. If X ⊆ V (G), then the set of neighbors of X is N (X) = ∪ u∈X N (u), G[X] is the subgraph induced by X, and X c is the complement of the subset X. For sets A, B ⊆ V (G), we use A \ B to denote the vertices belonging to A but not B. For such disjoint A and B we let (A, B) denote the set of edges such that each edge is incident to both a vertex in A and a vertex in B.
A matching M is a set of pairwise non-incident edges of G. A matching of maximum cardinality is a maximum matching and µ(G) is the cardinality of such a maximum matching. For a set A ⊆ V (G) and matching M , we say A is saturated by M if every vertex of A is incident to an edge in M . For two disjoint sets A, B ⊆ V (G), we say there is a matching M of A into B if M is a matching of G such that every edge of M belongs to (A, B) and each vertex of A is saturated. An M -alternating path is a path that alternates between edges in M and those not in M . An M -augmenting path is an M -alternating path which begins and ends with an edge not in M .
A set S ⊆ V (G) is independent if no two vertices from S are adjacent. An independent set of maximum cardinality is a maximum independent set and α(G) is the cardinality of such a maximum independent set. For a graph G, let Ω(G) denote the family of all its maximum independent sets, let See [1,9,14] for background and properties of core(G) and corona(G). For a graph G and a set X Zhang [16] showed that max{d(X) : The set S ⊆ V (G) a critical independent set if S is both a critical set and independent. A critical independent set of maximum cardinality is called a maximum critical independent set. Note that for some graphs the empty set is the only critical independent set, for example odd cycles or complete graphs. See [2,7,8,16] for more background and properties of critical independent sets.
Finding a maximum independent set is a well-known NP-hard problem. Zhang [16] first showed that a critical independent set can be found in polynomial time. Butenko and Trukhanov [2] showed that every critical independent set is contained in a maximum independent set, thereby directly connecting the problem of finding a critical independent set to that of finding a maximum independent set.
For a graph G the inequality α(G)+µ(G) ≤ n always holds. A graph G is a König-Egerváry graph if α(G) + µ(G) = n. All bipartite graphs are König-Egerváry but there are non-bipartite graphs which are König-Egerváry as well, see Figure 2 for an example. We adopt the convention that the empty graph K 0 , without vertices, is a König-Egerváry graph. In [7] it was shown that König-Egerváry graphs are closely related to critical independent sets. Theorem 1.1. [7] A graph G is König-Egerváry if, and only if, every maximum independent set in G is critical.
For any graph G, there is a unique set X ⊆ V (G) such that all of the following hold: is a König-Egerváry graph, (iii) for every non-empty independent set S in G[X c ], |N (S)| ≥ |S|, and (iv) for every maximum critical indendent set I of G, X = I ∪ N (I).
Larson in [8] showed that a maximum critical independent set can be found in polynomial time. So the decomposition in Theorem 1.2 of a graph G into X and X c is also computable in polynomial time. Figure 1 gives an example of this decomposition, where both the sets X and X c are non-empty. Recall, for some graphs the empty set is the only critical independent set, so for such graphs the set X would be empty. If a graph G is a König-Egerváry graph, then the set X c would be empty. We adopt the convention that if K 0 is empty graph, then α(K 0 ) = 0. In [5,11] the following concepts were introduced: for a graph G, However, the following result due to Larson allows us to use a more suitable definition for diadem(G).

Theorem 1.3. [8]
Each critical independent set is contained in some maximum critical independent set.
For the remainder of this paper we define diadem(G) = {S : S is a maximum critical independent set in G}.
Note that if G is a graph where the empty set is the only critical indepedent set (including the case G = K 0 , the empty graph), then ker(G), diadem(G), and nucleus(G) are all empty. See Figure 2 for examples of the sets ker(G), diadem(G), and nucleus(G).
is not a König-Egerváry graph and has ker( In [4,5], the following necessary conditions for König-Egerváry graphs were given: In [4] it was conjectured that condition (i) of Theorem 1.4 is sufficient for König-Egerváry graphs and in [5] it was conjectured the necessary condition in Theorem 1.5 is also sufficient. The purpose of this paper is to affirm these conjectures by proving the following new characterizations of König-Egerváry graphs.
Theorem 1.6. For a graph G, the following are equivalent: The paper [4] gives an upper bound for α(G) in terms of unions and intersections of maximum independent sets, proving 2α(G) ≤ | core(G)| + | corona(G)| for any graph G. It is natural to ask whether a similar lower bound for α(G) can be formulated in terms of unions and intersections of critical independent sets. Jarden, Levit, and Mandrescu in [4] conjectured that for any graph G, the inequality | ker(G)| + | diadem(G)| ≤ 2α(G) always holds. We will prove a slightly stronger statement. By Theorem 1.3 we see that ker(G) ⊆ nucleus(G) holds implying that | ker(G)| + | diadem(G)| ≤ | nucleus(G)| + | diadem(G)|. In section 4 we will prove the following statement, resolving the cited conjecture: It would be interesting to know whether the sets nucleus(G) and diadem(G), or their sizes, can be computed in polynomial time.

Some structural lemmas
Here we prove several crucial lemmas which will be needed in our proofs. Our results hinge upon the structure of the set X as described in Theorem 1.2.
Lemma 2.1. Let I be a maximum critical independent set in G and set X = I ∪ N (I). Then diadem(G) ∪ N (diadem(G)) = X.
Proof. By Theorem 1.2 the set X is unique in G, that is, for any maximum critical independent set S, X = S ∪ N (S). Then diadem(G) = X follows by definition. Proof. Let S be a maximum critical independent set in G. Using Theorem 1.2 we see that S is a maximum independent set in G[X] and also G[X] is a König-Egerváry graph. Then Theorem 1.1 gives that S must also be critical in G[X], which implies that diadem(G) ⊆ diadem(G[X]). Now let v ∈ nucleus(G[X]). Then v belongs to every maximum critical indepedent set in G[X]. As remarked above, since every maximum critical independent set in G is also a maximum critical independent set in G[X], then v belongs to every maximum critical independent set in G. This shows that v ∈ nucleus(G) and nucleus(G[X]) ⊆ nucleus(G) follows.
is an independent set in G[X] larger than S, which cannot happen. Therefore we must have |S ′ | ≥ |A ′ | as desired. Proof. First note that if the set X is empty, then by Lemma 2.1 both sides of the inequality are zero. So let us assume that X is non-empty. Now consider the set A = nucleus(G) \ nucleus(G[X]). If this independent set is empty, then nucleus(G) = nucleus(G[X]) and there is nothing to prove since diadem(G) ⊆ diadem(G[X]) holds by Lemma 2.2. If A is non-empty, for each v ∈ A there is some maximum independent set S of G[X] which doesn't contain v. Since S is a maximum independent set there exists u ∈ N (v) ∩ S.
Since v ∈ nucleus(G), then u does not belong to any maximum critical independent set in G. Recall by Theorem 1.2 (ii) G[X] is a König-Egerváry graph, so Theorem 1.1 gives that S is a maximum critical independent set in G[X]. It follows that u ∈ diadem(G[X]) \ diadem(G), which shows each vertex in A is adjacent to at least one vertex in diadem(G[X]) \ diadem(G). Now we will show there is a maximum matching from A into diadem(G[X])\ diadem(G) with size |A|. For sake of contradiction, suppose such a matching M has less than |A| edges. Then there exists some vertex v ∈ A not saturated by M . By the above, v is adjacent to some vertex u ∈ diadem(G[X]) \ diadem(G). Since M is maximum, u is matched to some vertex w ∈ A under M . Now let S be a maximum independent set of G[X] containing u. We now restrict ourselves to the subgraph induced by the edges (A ∩ N (S), S ∩ N (A)), noting this subgraph is bipartite since both A ∩ N (S) and S ∩ N (A) are independent. In this subgraph, consider the set P of all M -alternating paths starting with the edge vu. Note that all such paths must start with the vertices v, u, then w. Also, such paths must end at either a matched vertex in A∩ N (S) or an unmatched vertex in S ∩ N (A).
We wish to show that there is some alternating path ending at an unmatched vertex in S ∩ N (A). For sake of contradiction, suppose all alternating paths end at a matched vertex in A ∩ N (S) and let V (P) denote the union of all vertices belonging to such an alternating path. We aim to show this scenario contradicts Lemma 2.3. Now clearly we must have N (V (P) ∩ A) ∩ S ⊆ V (P) ∩ S, else we could extend an alternating path to any vertex in (N (V (P) ∩ A) ∩ S) \ (V (P) ∩ S). Also, since all paths in P end at a matched vertex in A ∩ N (S), then every vertex of V (P) ∩ S is matched under M , and such a situation should look as in Figure 3. From this it follows that |V (P) ∩ S| < |V (P) ∩ A|. The previous statements exactly contradict Lemma 2.3, so there is some alternating path P ending at an unmatched vertex x ∈ S ∩N (A). This means that P is an M -augmenting path. A well-known theorem in graph theory states that a matching is maximum in G if, and only if, there is no augmenting path [15]. So P being an M -augmenting path contradicts our assumption that M is a maximum matching. Therefore there is a matching M from A into diadem (

New characterizations of König-Egerváry graphs
Proof (of Theorem 1.6). First we prove (ii) ⇒ (i). Suppose that diadem(G) = corona(G) holds and let I be a maximum critical independent set with X = I ∪ N (I). We will use the decomposition in Theorem 1.2 to show that X c must be empty and hence, G = G[X] is a König-Egerváry graph. By Lemma 2.1 we have corona(G) = diadem(G) ⊆ X, in other words every maximum independent set in G is contained in X. This implies that |I| = α(G[X]) = α(G). Now by Theorem 1.
showing that we must have α(G[X c ]) = 0. Now clearly the result follows, since α(G[X c ]) = 0 implies that X c must be empty.
To prove (iii) ⇒ (i), again we will use the decomposition in Theorem 1.2 to show that X c must be empty and hence, G is a König-Egerváry graph. So suppose that | diadem(G)| + | nucleus(G)| = 2α(G) and let I be a maximum critical independent set in G with X = I ∪ N (I). Lemma 2.4 implies that  Combining Theorem 1.7 and the inequality 2α(G) ≤ | core(G)|+| corona(G)| proven in [4], the following corollary is immediate. These upper and lower bounds are quite interesting. The fact that every critical independent set is contained in a maximum independent set implies that diadem(G) ⊆ corona(G) for all graphs G. However, the graph G 2 in Figure 2 has core(G 2 ) nucleus(G 2 ) while the graph G in Figure 1 has nucleus(G) = {a, b, c} core(G) = {a, b, c, h}.

Acknowledgements
Many thanks to my advisor László Székely for feedback on initial versions of this manuscript. Partial support from the NSF DMS under contract 1300547 is gratefully acknowledged.