Critical and maximum independent sets of a graph (original) (raw)

Critical Independent Sets of a Graph

2014

Let G be a simple graph with vertex set V (G). A set S ⊆ V (G) is independent if no two vertices from S are adjacent, and by Ind(G) we mean the family of all independent sets of G. The number d (X) = |X| − |N (X)| is the difference of X ⊆ V (G), and a set A ∈ Ind(G) is critical if d(A) = max{d (I) : I ∈ Ind(G)} [26]. Let us recall the following definitions: core (G) = {S : S is a maximum independent set } [10], corona (G) = {S : S is a maximum independent set } [2], ker(G) = {S : S is a critical independent set } [12], diadem(G) = {S : S is a critical independent set }. In this paper we present various structural properties of ker(G), in relation with core (G), corona (G), and diadem(G).

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

Computing Research Repository, 2011

Let G=(V,E). A set S is independent if no two vertices from S are adjacent. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set}. Let us recall that ker(G) is the intersection of all critical independent sets, and core(G) is the intersection of all maximum independent sets. Recently, it was established that ker(G) is a subset of core(G) is true for every graph, while the corresponding equality holds for bipartite graphs. In this paper we present various structural properties of ker(G). The main finding claims that ker(G) is equal to the union of all inclusion minimal independent sets with positive difference.

Critical independent sets and König–Egerváry graphs

Graphs and Combinatorics, 2009

A set S of vertices is independent in a graph G, and we write S ∈ Ind(G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while core(G) denotes the intersection of all maximum independent sets .

On König-Egerváry collections of maximum critical independent sets

The Art of Discrete and Applied Mathematics

Let G be a simple graph with vertex set V (G). A set S ⊆ V (G) is independent if no two vertices from S are adjacent. The graph G is known to be a König-Egerváry if α (G) + µ (G) = |V (G)|, where α (G) denotes the size of a maximum independent set and µ (G) is the cardinality of a maximum matching. The number d (X) = |X| − |N (X)| is the difference of X ⊆ V (G), and a set A ∈ Ind(G) is critical if d(A) = max{d (I) : I ∈ Ind(G)} [21]. Let Ω(G) denote the family of all maximum independent sets. Let us say that a family Γ ⊆ Ind(G) is a König-Egerváry collection if | Γ| + | Γ| = 2α(G) [5]. In this paper, we show that if the family of all maximum critical independent sets of a graph G is a König-Egerváry collection, then G is a König-Egerváry graph. It generalizes one of our conjectures recently validated in [19].

On Minimal Critical Independent Sets of Almost Bipartite non-Konig-Egervary Graphs

Cornell University - arXiv, 2022

A set S ⊆ V is independent in a graph G = (V, E) if no two vertices from S are adjacent. The independence number α(G) is the cardinality of a maximum independent set, while µ(G) is the size of a maximum matching in G. If α(G)+µ(G) equals the order of G, then G is called a König-Egerváry graph [6, 25]. The number d (G) = max{|A| − |N (A)| : A ⊆ V } is called the critical difference of G [27] (where N (A) = {v : v ∈ V, N (v) ∩ A = ∅}). It is known that α(G) − µ(G) ≤ d (G) holds for every graph [16, 23, 24]. A graph G is (i) unicyclic if it has a unique cycle, (ii) almost bipartite if it has only one odd cycle. Let ker(G) = {S : S is a critical independent set }, core(G) be the intersection of all maximum independent sets, and corona(G) be the union of all maximum independent sets of G. It is known that ker(G) ⊆ core(G) is true for every graph [16], while the equality holds for bipartite graphs [19], and for unicyclic non-König-Egerváry graphs [20]. In this paper, we prove that if G is an almost bipartite non-König-Egerváry graph, then ker(G) = core(G), corona(G) ∪ N (core(G)) = V (G), and |corona(G)|+ |core(G)| = 2α(G) + 1.

K-Independence Critical Graphs

Australas. J Comb., 2012

Let k be a positive integer and G = (V (G), E(G)) a graph. A subset S of V (G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k − 1. The maximum cardinality of a k-independent set of G is the k-independence number β k (G). In this paper, we study the properties of graphs for which the k-independence number changes whenever an edge or vertex is removed or an edge is added.

Maximal k-independent sets in graphs

Discussiones Mathematicae Graph Theory, 2008

A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted i k (G) and β k (G). We give some relations between β k (G) and β j (G) and between i k (G) and i j (G) for j = k. We study two families of extremal graphs for the inequality i 2 (G) ≤ i(G) + β(G). Finally we give an upper bound on i 2 (G) and a lower bound when G is a cactus.

On the Intersection of All Critical Sets of a Unicyclic Graph

2011

A set S is independent in a graph G if no two vertices from S are adjacent. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. If alpha(G)+mu(G)=|V|, then G=(V,E) is called a Konig-Egervary graph. The number d_{c}(G)=max{|A|-|N(A)|} is called the critical difference of G (Zhang, 1990). By core(G) (corona(G)) we denote the intersection (union, respectively) of all maximum independent sets, while by ker(G) we mean the intersection of all critical independent sets. A connected graph having only one cycle is called unicyclic. It is known that ker(G) is a subset of core(G) for every graph G, while the equality is true for bipartite graphs (Levit and Mandrescu, 2011). For Konig-Egervary unicyclic graphs, the difference |core(G)|-|ker(G)| may equal any non-negative integer. In this paper we prove that if G is a non-Konig-Egervary unicyclic graph, then: (i) ker(G)= core(G) and (ii) |corona(G)|+|core(G)|=2*alpha(G)+1. Pay attention that |corona(G)|+|core(G)|=2*alpha(G) holds for every Konig-Egervary graph.

Monotonic Properties of Collections of Maximum Independent Sets of a Graph

Order

Let G be a simple graph with vertex set V (G). A set S ⊆ V (G) is independent if no two vertices from S are adjacent. The graph G is known to be a König-Egerváry if α (G) + µ (G) = |V (G)|, where α (G) denotes the size of a maximum independent set and µ (G) is the cardinality of a maximum matching. Let Ω(G) denote the family of all maximum independent sets, and f be the function from subcollections Γ of Ω(G) to N such that f (Γ) = | Γ| + | Γ|. Our main finding claims that f is ⊳-increasing, where the preorder Γ ′ ⊳ Γ means that Γ ′ ⊆ Γ and Γ ⊆ Γ ′. Let us say that a family ∅ = Γ ⊆ Ω (G) is a König-Egerváry collection if | Γ| + | Γ| = 2α(G). We conclude with the observation that for every graph G each subcollection of a König-Egerváry collection is König-Egerváry as well.