A Study on Integer Additive Set-Graceful 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 $f^{\oplus}:E(G)\rightarrow \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective. An integer additive set-labeling is an injective function $f:V(G)\rightarrow \mathcal{P}(\mathbb{N}_0)$, $\mathbb{N}_0$ is the set of all non-negative integers and an integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \rightarrow \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is also injective. In this paper, we extend the concepts of set-graceful labeling to integer additive set-labelings of graphs and provide some results on them.Comment: 11 pages, submitted to JARP

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

An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles

In 1991, Gnanajothi [4] proved that the path graph P_n with n vertex and n-1 edge is odd graceful, and the cycle graph C_m with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd. In this paper, firstly, we studied the graph C_m $\cup$ P_m when m = 4, 6,8,10 and then we proved that the graph C_ $\cup$ P_n is odd graceful if m is even. Finally, we described an algorithm to label the vertices and the edges of the vertex set V(C_m $\cup$ P_n) and the edge set E(C_m $\cup$ P_n).Comment: 9 Pages, JGraph-Hoc Journa

Optimal Dynamic Distributed MIS

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a $3$-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.Comment: 19 pages including appendix and reference

On Integer Additive Set-Indexers of Graphs

A set-indexer of a graph $G$ is an injective set-valued function $f:V(G) \rightarrow2^{X}$ such that the function $f^{\oplus}:E(G)\rightarrow2^{X}-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus} f(v)$ for every $uv{\in} E(G)$ is also injective, where $2^{X}$ is the set of all subsets of $X$ and $\oplus$ is the symmetric difference of sets. An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. A graph $G$ which admits an IASI is called an IASI graph. An IASI $f$ is said to be a {\em weak IASI} if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ and an IASI $f$ is said to be a {\em strong IASI} if $|g_f(uv)|=|f(u)| |f(v)|$ for all $u,v\in V(G)$. In this paper, we study about certain characteristics of inter additive set-indexers.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1312.7674 To Appear in Int. J. Math. Sci.& Engg. Appl. in March 201