Purnima Gupta

Domination in graphoidal covers of a graph

Abstract A graphoidal cover of a given graph G =( V , E ) is a set of its paths of length at least one, not necessarily open, such that no two paths have a common internal vertex and every edge of G is in exactly one of these paths. Graphoidal covers provide a fresh ground for generalizing results in graph theory and this paper is the first attempt to demonstrate the fruitfulness of this contention taking the notion of domination in graphs. Given a graphoidal cover ψ of G we define a set D of vertices of G to be a ψ - dominating set ( ψ - domset , for short) of G whenever for every vertex v in V ⧹ D there exists a vertex u in D and a path P in ψ such that u and v are the end-vertices of P. This paper initiates a study of this concept in graphs which may not be necessarily finite.

On point-set domination in graphs IV: separable graphs with unique minimum psd-sets

A set D of vertices in a graph G = (V,E) is called a point-set dominating (or, psd-) set of G if for every nonempty subset S of V − D there exists v ϵ D such that the induced subgraph 〈S ∪ {v}〉 is connected (cf. Sampthkumar and Pushpa Latha (1993) [6]). Here, we report results of our investigation into the nature of connected separable graphs having unique minimum psd-sets. In particular, we characterize block-cactus graphs (with at least two blocks) having this property.

Domination in Graphoidally Covered Graphs: Least-Kernel Graphoidal Covers

Abstract Given a graph G = ( V , E ) (not necessarily finite), a graphoidal cover of G means a collection Ψ of non-trivial paths in G called Ψ-edges, which are not necessarily open (not necessarily finite), such that every vertex of G is an internal vertex of at most one path in Ψ and every edge of G is in exactly one path in Ψ. The set of all graphoidal covers of a graph G = ( V , E ) is denoted by G G and for a given Ψ ∈ G G the ordered pair (G, Ψ) is called a graphoidally covered graph. Two vertices u and v of G are Ψ-adjacent if they are the ends of an open Ψ-edge. A set D of vertices in (G, Ψ) is Ψ-independent if no two vertices in D are Ψ-adjacent and is said to be Ψ-dominating if every vertex of G is either in D or is Ψ-adjacent to a vertex in D; G is γ Ψ ( G ) -definable ( γ i Ψ ( G ) -definable) if (G, Ψ) has a finite Ψ-dominating (Ψ-independent Ψ-dominating) set. Clearly, if G is γ i Ψ ( G ) -definable, then G is γ Ψ ( G ) -definable and γ Ψ ( G ) ≤ γ i Ψ ( G ) . Further if for a graphoidal cover Ψ of G, γ Ψ ( G ) = γ i Ψ ( G ) then we call Ψ as a least-kernel graphoidal cover of G (in short, an LKG cover of G). A natural question arises: “Does every graph possess an LKG cover?” We firstly exhibit that not every graph possesses an LKG cover and thereafter, in the quest to find graphs possessing an LKG cover, we proved that every finite tree and every finite unicyclic graph admits an LKG cover. We further extend Allan Laskar theorem to infinite graphs by showing every γ ( G ) -definable infinite claw free graph has an LKG cover.

Bounds on Graphoidal Length of a Graph

Abstract A graphoidal cover of a graph G is a set Ψ of non-trivial paths (which are not necessarily open) in G such that every vertex of G is an internal vertex of at most one path in Ψ and every edge of G is in exactly one path in Ψ. We denote the set of all graphoidal covers of graph G by G G . In this paper we introduce a parameter gl(G), called graphoidal length of the graph G and is defined as g l ( G ) = max Ψ ∈ G G ⁡ { min P ∈ Ψ ⁡ l ( P ) } . We give bounds for the parameter gl(G) in terms of the well known and well studied parameter η ( G ) , graphoidal covering number of the graph and show that the bounds are sharp.

On graphs whose domination numbers equal their independent domination numbers

Abstract In this paper, we extend a result due to R. B. Allan and R. C. Laskar on graphs whose independent domination numbers equal their domination numbers. We will consider finite simple graphs as treated in most of the standard text-books on Graph Theory (e.g., see D. B. West [1]). Let G = (V,E) be any graph and D ⊆ V. We let N(D) denote the set of all vertices adjacent to those in D; the vertices in N(D) are often called the neighbours of those in D. In particular, if D = {u} then N(D) is written simply as N(u) The set D is called a dominating set (or, simply a domset) of G if N(ν) ∩ D ≠ ∅ for every ν ∈ V - D and an independent domset is often called a kernel of G (see [2, 3, 4, 5]). We will denote by D(G) (Di(G)) the set of all domsets (kernels) in G If G is a finite graph then the least cardinality of a domset (kernel) of G is called its domination (independent domination) number, denoted γ(G) (γi(G)). The following result is well known. We thank the referees for pointing out an error in the original version of the paper which led to substantial improvement in the main result of this note.

Bounds on 2-point set domination number of a graph

A set D ⊆ V (G) is a 2-point set dominating set (2-psd set) of a graph G if for any subset S ⊆ V − D, there exists a nonempty subset T ⊆ D containing at most two vertices such that the subgraph 〈S ∪ T〉 induced by S ∪ T is connected. The 2-point set domination number of G, denoted by γ2ps(G), is the minimum cardinality of a 2-psd set of G. In this paper, we determine the lower bounds and an upper bound on γ2ps(G) of a graph. We also characterize extremal graphs for the lower bounds and identify some well-known classes of both separable and nonseparable graphs attaining the upper bound.

Graphoidal Length and Graphoidal Covering Number of a Graph

Let \(G=(V,E)\) be a finite graph. A graphoidal cover \(\varPsi \) of G is a collection of paths (not necessary open) in G such that every vertex of G is an internal vertex of at most one path in \(\varPsi \) and every edge of G is in exactly one path in \(\varPsi .\) The graphoidal covering number \(\eta \) of G is the minimum cardinality of a graphoidal cover of G. The length \(gl_{\varPsi }(G)\) of a graphoidal cover \(\varPsi \) of G is defined to be \(\min \{l(P): P\in \varPsi \}\) where l(P) is the length of the path P. The graphoidal length gl(G) is defined to be \(\max \{gl_{\varPsi }(G): \varPsi \) is a graphoidal cover of \(G\}.\) In this paper we investigate the existence of graphs which admit a graphoidal cover \(\varPsi \) with \(|\varPsi |=\eta (G)\) and \(gl_{\varPsi }(G)=gl(G)\).

Global 2-Point Set Domination Number of a Graph

Abstract A set D of vertices in a graph G = ( V , E ) is called a 2-point set dominating set of G if for every set T ⊆ V − D there exists a non-empty set S ⊆ D containing at most two vertices such that the induced subgraph 〈 S ∪ T 〉 is connected. A set D ⊆ V ( G ) is called a global 2-point set dominating set of G if D is a 2-point set dominating set of both G and G ‾ . The global 2-point set domination number (2-point set domination number) is the minimum cardinality of a global 2-point set dominating set (2-point set dominating set) in G. In this paper we determine bounds on the global 2-point set domination number of a graph in terms of other graph invariants. We have also given relation between global 2-point set domination number and 2-point set domination number for some classes of graphs.

Zero Ring Labeling of Graphs

Abstract This paper introduces the notion of zero ring labeling of a graph and its empirical study demonstrates that every graph admits a zero ring labeling with respect to some zero ring. The zero ring graph Γ ( R 0 ) turns out to be maximal with respect to an injective zero ring labeling. In particular, we determine the optimal zero ring index for some well-known graphs.

Finite Abelian Group Labeling

Abstract In this paper, we introduce an abelian group labeling (shortly, AGL) over finite abelian groups. We have shown that every finite graph admits an abelian group labeling. In the course of investigation, we found that representation labeling can be obtained from abelian group labeling for certain graphs. Several new directions for further research are also indicated through problems.