A Study on Topological Integer Additive Set-Labeling of Graphs

A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. Let $G$ be a graph and let $X$ be a non-empty set. A set-indexer $f:V(G)\to \mathcal{P}(X)$ is called a topological set-labeling of $G$ if $f(V(G))$ is a topology of $X$. An integer additive set-labeling is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$, whose associated function $f^+:E(G)\to \mathcal{P}(\mathbb{N}_0)$ is defined by $f(uv)=f(u)+f(v), uv\in E(G)$, where $\mathbb{N}_0$ is the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ is its power set. An integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of topological set-labeling of graphs to topological integer additive set-labeling of graphs.Comment: 16 pages, 7 figures, Accepted for publication. arXiv admin note: text overlap with arXiv:1403.398

Topological Integer Additive Set-Sequential Graphs

Let $\mathbb{N}_0$ denote the set of all non-negative integers and $X$ be any non-empty subset of $\mathbb{N}_0$. Denote the power set of $X$ by $\mathcal{P}(X)$. An integer additive set-labeling (IASL) of a graph $G$ is an injective set-valued function $f:V(G)\to \mathcal{P}(X)$ such that the induced function $f^+:E(G) \to \mathcal{P}(X)$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sumset of $f(u)$ and $f(v)$. If the associated set-valued edge function $f^+$ is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL $f$ is said to be a topological IASL (TIASL) if $f(V(G))\cup \{\emptyset\}$ is a topology of the ground set $X$. An IASL is said to be an integer additive set-sequential labeling (IASSL) if $f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}$. An IASL of a given graph $G$ is said to be a topological integer additive set-sequential labeling of $G$, if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of $G$. In this paper, we study the conditions required for a graph $G$ to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs.Comment: 10 pages, 2 figures. arXiv admin note: text overlap with arXiv:1506.0124

Topological Integer Additive Set-Valued Graphs: A Review

International audienceLet í µí±‹ denote a set of non-negative integers and í µí±ƒ 0 (í µí±‹) be the collection of all non-empty subsets of í µí±‹. An integer additive set-labeling (IASL) of a graph í µí°º is an injective set-valued function í µí±“: í µí±‰(í µí°º) → í µí±ƒ 0 (í µí±‹) where induced function í µí±“ + : í µí°¸(í µí°º) → í µí±ƒ 0 (í µí±‹) is defined byí µí±“ + (í µí±¢í µí±£) = í µí±“(í µí±¢) + í µí±“(í µí±£), where í µí±“(í µí±¢) + í µí±“(í µí±£) is the sumset of í µí±“(í µí±¢) and í µí±“(í µí±£). A set-labeling í µí±“: í µí±‰(í µí°º) → í µí±ƒ 0 (í µí±‹) is said to be a topological set-labeling if í µí±“(í µí±‰(í µí°º)) ∪ {∅} is a topology on the ground set í µí±‹ and a set-labeling í µí±“: í µí±‰(í µí°º) → í µí±ƒ 0 (í µí±‹) is said to be a topogenic set-labeling if í µí±“(í µí±‰(í µí°º)) ∪ í µí±“ + (í µí°¸(í µí°º)) ∪ {∅} is a topology on í µí±‹. In this article, we critically review some interesting studies on the properties and characteristics of different topological and topogenic integer additive set-labeling of certain graphs

A Study on Topological Integer Additive Set-Labeling of Graphs

A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$, where $X$ is a finite set and a set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. Let $G$ be a graph and let $X$ be a non-empty set. A set-indexer $f:V(G)\to \mathcal{P}(X)$ is called a topological set-labeling of $G$ if $f(V(G))$ is a topology of $X$. An integer additive set-labeling is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$, whose associated function $f^+:E(G)\to \mathcal{P}(\mathbb{N}_0)$ is defined by $f(uv)=f(u)+f(v), uv\in E(G)$, where $\mathbb{N}_0$ is the set of all non-negative integers and $\mathcal{P}(\mathbb{N}_0)$ is its power set. An integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of topological set-labeling of graphs to topological integer additive set-labeling of graphs

On Integer Additive Set-Valuations of Finite Jaco Graphs

International audienceLet X denote a set of non-negative integers and P (X ) be its power set. An integer additive set-labeling (IASL) of a graph G is an injective set-valued function f : V (G) → P (X ) −{;} where induced function f + : E(G) → P (X ) − {;} is defined by f + (uv) = f (u) + f (v), where f (u) + f (v) is the sumset of f (u) and f (v). Let f (x) = mx + c; m ∈ N , c ∈ N 0 . A finite linear Jaco graph, denoted by J n ( f (x)), is a directed graph with vertex set { v i : i ∈ N } such that (v i , v j ) is an arc of J n ( f (x)) if and only if f (i) + i − d − (v j ) ≥ j. In this paper, we discuss the admissibility of different types of integer additive set-labeling by finite linear Jaco graphs

A study on topogenic integer additive set-labeled graphs

International audienceLet N 0 denote the set of all non-negative integers and P(N 0) be its power set. An integer additive set-labeling (IASL) of a graph G is an injective function f : V (G) → P(N 0) such that the induced function f + : E(G) → P(N 0) is defined by f + (uv) = f (u) + f (v), where f (u) + f (v) is the sumset of f (u) and f (v). The IASL f is said to be an integer additive set-indexer (IASI) if the associated function f + is also injective. If f + (uv) = k ∀ uv ∈ E(G), then f is said to be a k-uniform integer additive set-indexer. In this paper, we study the admissibility of a particular type of integer additive set-labeling, called topogenic IASL, by certain graphs

Labeling Schemes with Queries

We study the question of ``how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels''. Let $f$ be a function on pairs of vertices. An $f$-labeling scheme for a family of graphs \cF labels the vertices of all graphs in \cF such that for every graph G\in\cF and every two vertices $u,v\in G$, the value $f(u,v)$ can be inferred by merely inspecting the labels of $u$ and $v$. This paper introduces a natural generalization: the notion of $f$-labeling schemes with queries, in which the value $f(u,v)$ can be inferred by inspecting not only the labels of $u$ and $v$ but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one {\em query}) enables us to reduce the label size of many labeling schemes significantly