A CORRECTION TO A PAPER ON ROMAN κ-DOMINATION IN GRAPHS

Title & Authors
A CORRECTION TO A PAPER ON ROMAN κ-DOMINATION IN GRAPHS
Mojdeh, Doost Ali; Moghaddam, Seyed Mehdi Hosseini;

Abstract
Let G = (V, E) be a graph and k be a positive integer. A $\small{k}$-dominating set of G is a subset $\small{S{\subseteq}V}$ such that each vertex in $\small{V{\backslash}S}$ has at least $\small{k}$ neighbors in S. A Roman $\small{k}$-dominating function on G is a function $\small{f}$ : V $\small{{\rightarrow}}$ {0, 1, 2} such that every vertex $\small{{\upsilon}}$ with $\small{f({\upsilon})}$ = 0 is adjacent to at least $\small{k}$ vertices $\small{{\upsilon}_1}$, $\small{{\upsilon}_2}$, $\small{{\ldots}}$, $\small{{\upsilon}_k}$ with $\small{f({\upsilon}_i)}$ = 2 for $\small{i}$ = 1, 2, $\small{{\ldots}}$, $\small{k}$. In the paper titled "Roman $\small{k}$-domination in graphs" (J. Korean Math. Soc. 46 (2009), no. 6, 1309-1318) K. Kammerling and L. Volkmann showed that for any graph G with $\small{n}$ vertices, $\small{{{\gamma}_{kR}}(G)+{{\gamma}_{kR}(\bar{G})}{\geq}}$ min $\small{\{2n,4k+1\}}$, and the equality holds if and only if $\small{n{\leq}2k}$ or $\small{k{\geq}2}$ and $\small{n=2k+1}$ or $\small{k=1}$ and G or $\small{\bar{G}}$ has a vertex of degree $\small{n}$ - 1 and its complement has a vertex of degree $\small{n}$ - 2. In this paper we find a counterexample of Kammerling and Volkmann's result and then give a correction to the result.
Keywords
dominating set;Roman k-dominating function;correction;
Language
English
Cited by
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