Common Independence in Graphs

Results in Mathematics, 1990

A set S of vertices of a graph G is dominating if each vertex z not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number,-y(G),

Discrete Mathematics, 2005

The average lower independence number i av (G) of a graph G=(V , E) is defined as 1 |V | v∈V i v (G), and the average lower domination number av (G) is defined as 1

Discrete Applied Mathematics, 2013

Let G be a graph and let k and j be positive integers. A subset D of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination

Discrete Mathematics, 2013

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we offer a survey of selected recent results on independent domination in graphs.

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.

Discrete Mathematics, 2003

Vertices of the independence graph of a graph G represent maximum independent sets of G, two vertices being adjacent whenever the corresponding sets are disjoint. Vizing's inequality involving the independence number of the Cartesian product of graphs G and H states that

European Journal of Pure and Applied Mathematics, 2021

A set S ⊆ V (G) is an independent transversal dominating set of a graph G if S is a dominating set of G and intersects every maximum independent set of G. An independent transversal dominating set which is a total dominating set is an independent transversal total dominating set. The minimum cardinality γit(G) (resp. γitt(G)) of an independent transversal dominating set (resp. independent transversal total dominating set) of G is the independent transversal domination number (resp. independent transversal total domination number) of G. In this paper, we show that for every positive integers a and b with 5 ≤ a ≤ b ≤ 2a − 2, there exists a connected graph G for which γit(G) = a and γitt(G) = b. We also study these two concepts in graphs which are the join, corona or composition of graphs.

Discrete Mathematics, 2001

For the independence number (G) of a connected graph G on n vertices with m edges the inequality (G)¿ 1 2 [(2m + n + 1) − (2m + n + 1) 2 − 4n 2 ] is proved and its algorithmic realization is discussed.

2018

Henning[6] introduced the concept of average domination and average independent domination. The domination number γv(G) of G relative to v is the minimum cardinality of a dominating set containing v. The average domination number of G is γav(G) = 1 |V (G)| ∑ v∈V (G) γv(G). The independent domination number iv(G) of G relative to v is the minimum cardinality of a maximal independent set in G that contains v. The average independent domination number of G is iav(G) = 1 |V (G)| ∑ v∈V (G) iv(G). In this note, we look at these parameters in a different point of view and hence simplify the results.

European Journal of Combinatorics, 1995