# Weak Set-Labeling Number of Certain IASL-Graphs

Source: [Weak Set-Labeling Number of Certain IASL-Graphs](<https://www.sota2.com/research/paper/68fa51a71e209131ce9759c0>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T11:32:06.782Z

Authors: N. K. Sudev, K. P. Chithra, K. A. Germina

Published: 2015-03-18T10:05:15.000Z

## Abstract

Let $\\mathbb{N}\_0$ be the set of all non-negative integers, let $X\\subset \\mathbb{N}\_0$ and $\\mathcal{P}(X)$ be the the power set of $X$. An integer additive set-labeling (IASL) of a graph $G$ is an injective function $f:V(G)\\to \\mathcal{P}(\\mathbb{N}\_0)$ such that the induced function $f^+:E(G) \\to \\mathcal{P}(\\mathbb{N}\_0)$ is defined by $f^+ (uv) = f(u)+ f(v)$, where $f(u)+f(v)$ is the sum set of $f(u)$ and $f(v)$. An IASL $f$ is said to be an integer additive set-indexer (IASI) of a graph $G$ if the induced edge function $f^+$ is also injective. An integer additive set-labeling $f$ is said to be a weak integer additive set-labeling (WIASL) if $\|f^+(uv)\|=\\max(\|f(u)\|,\|f(v)\|)~\\forall ~ uv\\in E(G)$. The minimum cardinality of the ground set $X$ required for a given graph $G$ to admit an IASL is called the set-labeling number of the graph. In this paper, we introduce the notion of the weak set-labeling number of a graph $G$ as the minimum cardinality of $X$ so that $G$ admits a WIASL with respect to the ground set $X$ and discuss the weak set-labeling number of certain graphs.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1503.07843>)

[PDF](<https://arxiv.org/pdf/1503.07843.pdf>)