On the Structure of the Minimum Critical Independent Set 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).

Critical and maximum independent sets of a graph

Discrete 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. By Ind(G) we mean the family of all independent sets of G, while core (G) and corona (G) denote the intersection and the union of all maximum independent sets, respectively. 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)} [23]. Let ker(G) and diadem(G) be the intersection and union, respectively, of all critical independent sets of G [13]. In this paper, we present various connections between critical unions and intersections of maximum independent sets of a graph. These relations give birth to new characterizations of König-Egerváry graphs, some of them involving ker(G), core (G), corona (G), and diadem(G).

Critical Sets in Bipartite Graphs

Arxiv preprint arXiv:1102.1138, 2011

Let G = (V, E) be a graph. A set S ⊆ V is independent if no two vertices from S are adjacent, and by Ind(G) (Ω(G)) we mean the set of all (maximum) independent sets of G, while α(G) = |S| for S ∈ Ω(G), and core(G) = ∩{S :

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.

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*alp...

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 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.

On unique independent sets in graphs

Discrete Mathematics, 1994

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