Fractional Domination of the Cartesian Products in Graphs 
Received:March 01, 2014 Revised:January 16, 2015 
Key Words:
Cartesian products fractional domination number fractional total domination number

Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.11361024; 11061014) and the Jiangxi Provincial Science and Technology Project (Grant No.KJLD12067). 

Hits: 2567 
Download times: 2482 
Abstract: 
Let $G=(V,E)$ be a simple graph. For any real function $g：V\longrightarrow R$ and a subset $S\subseteq V $, we write $g(S)=\sum_{v \in S}g(v)$. A function $f:V\longrightarrow [0,1]$ is said to be a fractional dominating function $(FDF)$ of $G$ if $f(N[v])\geq 1$ holds for every vertex $v\in V(G)$. The fractional domination number $\gamma_{f}(G)$ of $G$ is defined as $\gamma_{f}(G)=\min \{f(V)f$ is an $FDF$ of $G$ \}. The fractional total dominating function $f$ is defined just as the fractional dominating function, the difference being that $f(N(v))\geq 1$ instead of $f(N[v])\geq 1$. The fractional total domination number $\gamma_{f}^{0}(G)$ of $G$ is analogous. In this note we give the exact values of $\gamma_{f}(C_{m}\times P_{n})$ and $\gamma_{f}^{0}(C_{m}\times P_{n})$ for all integers $m\geq 3$ and $n\geq 2$. 
Citation: 
DOI:10.3770/j.issn:20952651.2015.03.005 
View Full Text View/Add Comment 