Graphs with unique maximum independent sets (original) (raw)

Common Independence in Graphs

Symmetry, 2021

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

A Note on Graphs Which Have Upper Irredundance Equal to Independence

Discrete Applied Mathematics, 1993

In this paper we consider the following graph parameters: IR(G), the upper irredundance number, Γ(G), the upper domination number and β(G), the independence number. It is well known that for any graph G, β(G)≤Γ(G)≤IR(G).We introduce the concept of a graph G being irredundant perfect ifIR(H)=β(H) for all induced subgraphs H of G. In this paper we characterize irredundant perfect graphs. This enables us to show that several classes of graphs are irredundant perfect, classes which include strongly perfect, bipartite and circular arc graphs.

Independent sets of some graphs

2013

Let G = (V, E) be a simple graph. A set S ⊆ V is independent set of G, if no two vertices of S are adjacent. The independence number α(G) is the size of a maximum independent set in the graph. Let R be a commutative ring with nonzero identity and I an ideal of R. The zero-divisor graph of R, denoted by Γ(R), is an undirected graph whose vertices are the nonzero zero-divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. Also the ideal-based zero-divisor graph of R, denoted by Γ I (R), is the graph which vertices are the set {x ∈ R\I|xy ∈ I for some y ∈ R\I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this paper we study the independent sets and the independence number of Γ(R) and Γ I (R).

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.

Independent Sets in Graphs without Subtrees with Many Leaves

—A subtree of a graph is called inscribed if no three vertices of the subtree generate a triangle in the graph. We prove that, for fixed k, the independent set problem is solvable in polynomial time for each of the following classes of graphs: (1) graphs without subtrees with k leaves, (2) subcubic graphs without inscribed subtrees with k leaves, and (3) graphs with degree not exceeding k and lacking induced subtrees with four leaves.

Independent [1, k]-sets in graphs

Australas. J Comb., 2014

We consider [1, k]-sets that are also independent, and note that not every graph has an independent [1, k]-set. For graphs having an independent [1, k]-set, we define the lower and upper [1, k]-independence numbers and determine upper bounds for these values. In addition, the trees having an independent [1, k]-set are characterized. Also, we show that the related decision problem is NP-complete.

On -independence in graphs with emphasis on trees

Discrete Mathematics, 2007

In a graph G = (V , E) of order n and maximum degree , a subset S of vertices is a k-independent set if the subgraph induced by S has maximum degree less or equal to k − 1. The lower k-independence number i k (G) is the minimum cardinality of a maximal k-independent set in G and the k-independence number k (G) is the maximum cardinality of a k-independent set. We show that i k n − + k − 1 for any graph and any k , and i 2 n − if G is connected, that k (T ) kn/(k + 1) for any tree, and that i 2 (n + s)/2 2 for any connected bipartite graph with s support vertices. Moreover, we characterize the trees satisfying i 2 = n − , k = kn/(k + 1), i 2 = (n + s)/2 or 2 = (n + s)/2. Lemma 2. For k 1, let w be a vertex of a graph G such that every neighbor of w has degree at most k, at least w or one of its neighbors has degree k or more, and every vertex in V (G ) \N [w], if any, has degree less than k in G . Let G be any graph and G the graph constructed from G and G by adding an edge between w and any vertex of G .

Maximum independent sets near the upper bound

Discrete Applied Mathematics, 2019

The size of a largest independent set of vertices in a given graph G is denoted by α(G) and is called its independence number (or stability number). Given a graph G and an integer K, it is NP-complete to decide whether α(G) ≥ K. An upper bound for the independence number α(G) of a given graph G with n vertices and m edges is given by α(G) ≤ p := 1 2 + 1 4 + n 2 − n − 2m. In this paper we will consider maximum independent sets near this upper bound. Our main result is the following: There exists an algorithm with time complexity O(n 2) that, given as an input a graph G with n vertices, m edges, p := 1 2 + 1 4 + n 2 − n − 2m , and an integer k ≥ 0 with p ≥ 2k + 1, returns an induced subgraph G p,k of G with n 0 ≤ p+2k+1 vertices such that α(G) ≤ p−k if and only if α(G p,k) ≤ p−k. Furthermore, we will show that we can decide in time O(1.2738 3k + kn) whether α(G p,k) ≤ p − k.

On independent [1,2]-sets in trees

Discussiones Mathematicae Graph Theory, 2018

An [1, k]-set S in a graph G is a dominating set such that every vertex not in S has at most k neighbors in it. If the additional requirement that the set must be independent is added, the existence of such sets is not guaranteed in every graph. In this paper we solve some problems previously posed by other authors about independent [1, 2]-sets. We provide a necessary condition for a graph to have an independent [1, 2]-set, in terms of spanning trees, and we prove that this condition is also sufficient for cactus graphs. We follow the concept of excellent tree and characterize the family of trees such that any vertex belongs to some independent [1, 2]-set. Finally, we describe a linear algorithm to decide whether a tree has an independent [1, 2]-set. This algorithm can be easily modified to obtain the cardinality of a smallest independent [1, 2]-set of a tree.

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).