Total irredundance in graphs (original) (raw)
Ramsey properties of generalised irredundant sets in graphs
Discrete Mathematics, 2001
For each vertex s of the subset S of vertices of a graph G, we deÿne Boolean variables p; q; r which measure the existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p; q; r) may be considered as a compound existence property of S-pns. The set S is called an f-set of G if f = 1 for all s ∈ S and the class of all f-sets of G is denoted by f . Special cases of f include the independent sets, irredundant sets and CO-irredundant sets of G. For some f ∈ F it is possible to deÿne analogues (involving f-sets) of the classical Ramsey graph numbers. We consider existence theorems for these f-Ramsey numbers and prove that some of them satisfy the well-known recurrence inequality which holds for the classical Ramsey numbers.
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.
Discrete Applied Mathematics, 1995
A vertex x in a subset X of vertices of a graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X-{x}. This paper describes the structure of bipartite graphs, chordal graphs and graphs of girth at least five in which every maxima1 set of vertices having no redundancies is maximum.
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 graphs with the third largest number of maximal independent sets
2009
Let G be a simple and undirected graph. By mi(G) we denote the number of maximal independent sets in G. Erdős and Moser posed the problem to determine the maximum cardinality of mi(G) among all graphs of order n and to characterize the corresponding extremal graphs attaining this maximum cardinality. The above problem has been solved by Moon and Moser in [J.W. Moon, L. Moser, On cliques in graphs, Israel J. Math. 3 (1965) 23-28]. More recently, Jin and Li [Z. Jin, X. Li, Graphs with the second largest number of maximal independent sets, Discrete Mathematics 308 (2008) 5864-5870] investigated the second largest cardinality of mi(G) among all graphs of order n and characterized the extremal graph attaining this value of mi(G). In this paper, we shall determine the third largest cardinality of mi(G) among all graphs G of order n. Additionally, graphs achieving this value are also determined.
A Note on the Sparing Number of Graphs
arXiv: Combinatorics, 2014
An integer additive set-indexer is defined as an injective function f:V(G)rightarrow2mathbbN0f:V(G)\rightarrow 2^{\mathbb{N}_0}f:V(G)rightarrow2mathbbN0 such that the induced function gf:E(G)rightarrow2mathbbN0g_f:E(G) \rightarrow 2^{\mathbb{N}_0}gf:E(G)rightarrow2mathbbN0 defined by gf(uv)=f(u)+f(v)g_f (uv) = f(u)+ f(v)gf(uv)=f(u)+f(v) is also injective. An IASI fff is said to be a weak IASI if ∣gf(uv)∣=max(∣f(u)∣,∣f(v)∣)|g_f(uv)|=max(|f(u)|,|f(v)|)∣gf(uv)∣=max(∣f(u)∣,∣f(v)∣) for all u,vinV(G)u,v\in V(G)u,vinV(G). A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph GGG, a vertex or an edge, is the cardinality of its set-labels. The sparing number of a graph GGG is the minimum number of edges with singleton set-labels, required for a graph GGG to admit a weak IASI. In this paper, we study the sparing number of certain graphs and the relation of sparing number with some other parameters like matching number, chromatic number, covering number, independence number etc.
Bull. ICA, 2020
If X is any nonempty set on n ≥ 2 elements we define the set graph Gn to be the graph whose vertices are the 2 − 2 proper subsets of X with two vertices adjacent if and only if their underlying sets are disjoint. We discuss some properties of Gn. In particular we find its clique partition number and its product dimension. We also give bounds for its representation number. We use standard graph theory terminology as given in [13]. A family of subsets S1, S2, . . . of a set S gives a graph in a natural way if we use these sets as vertices and let SiSj for i 6= j be an edge if and only if the corresponding subsets have a nonempty intersection. In [12], Marczewski has established the converse, i.e. for any graph G there is a set S, such that a family of its subsets defines G according to the above description. Erdős, Goodman and Posa in [1] have remarked that one may replace the idea of a nonempty intersection with disjointness of the subsets since the same would then imply Marczewski’s...