# Fractional global domination in graphs

• Volume: 30, Issue: 1, page 33-44
• ISSN: 2083-5892

top

## Abstract

top
Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g\left(N\left[v\right]\right)={\sum }_{u\in N\left[v\right]}g\left(u\right)\ge 1$ and $g\left(\overline{N\left(v\right)}\right)={\sum }_{u\notin N\left(v\right)}g\left(u\right)\ge 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number ${\gamma }_{fg}\left(G\right)$ is defined as follows: ${\gamma }_{fg}\left(G\right)$ = min|g|:g is an MGDF of G where $|g|={\sum }_{v\in V}g\left(v\right)$. In this paper we initiate a study of this parameter.

## How to cite

top

Subramanian Arumugam, Kalimuthu Karuppasamy, and Ismail Sahul Hamid. "Fractional global domination in graphs." Discussiones Mathematicae Graph Theory 30.1 (2010): 33-44. <http://eudml.org/doc/270977>.

@article{SubramanianArumugam2010,
abstract = {Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g(N[v]) = ∑_\{u ∈ N[v]\}g(u) ≥ 1$ and $g(\overline\{N(v)\}) = ∑_\{u ∉ N(v)\}g(u) ≥ 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number $γ_\{fg\}(G)$ is defined as follows: $γ_\{fg\}(G)$ = min|g|:g is an MGDF of G where $|g| = ∑_\{v ∈ V\} g(v)$. In this paper we initiate a study of this parameter.},
author = {Subramanian Arumugam, Kalimuthu Karuppasamy, Ismail Sahul Hamid},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination; global domination; dominating function; global dominating function; fractional global domination number},
language = {eng},
number = {1},
pages = {33-44},
title = {Fractional global domination in graphs},
url = {http://eudml.org/doc/270977},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Subramanian Arumugam
AU - Kalimuthu Karuppasamy
AU - Ismail Sahul Hamid
TI - Fractional global domination in graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 1
SP - 33
EP - 44
AB - Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g(N[v]) = ∑_{u ∈ N[v]}g(u) ≥ 1$ and $g(\overline{N(v)}) = ∑_{u ∉ N(v)}g(u) ≥ 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number $γ_{fg}(G)$ is defined as follows: $γ_{fg}(G)$ = min|g|:g is an MGDF of G where $|g| = ∑_{v ∈ V} g(v)$. In this paper we initiate a study of this parameter.
LA - eng
KW - domination; global domination; dominating function; global dominating function; fractional global domination number
UR - http://eudml.org/doc/270977
ER -

## References

top
1. [1] S. Arumugam and R. Kala, A note on global domination in graphs, Ars Combin. 93 (2009) 175-180. Zbl1224.05357
2. [2] S. Arumugam and K. Rejikumar, Basic minimal dominating functions, Utilitas Mathematica 77 (2008) 235-247. Zbl1160.05046
3. [3] G. Chartrand and L. Lesniak, Graphs & Digraphs (Fourth Edition, Chapman & Hall/CRC, 2005).
4. [4] E.J. Cockayne, G. MacGillivray and C.M. Mynhardt, Convexity of minimal dominating funcitons of trees-II, Discrete Math. 125 (1994) 137-146, doi: 10.1016/0012-365X(94)90154-6. Zbl0793.05117
5. [5] E.J. Cockayne, C.M. Mynhardt and B. Yu, Universal minimal total dominating functions in graphs, Networks 24 (1994) 83-90, doi: 10.1002/net.3230240205. Zbl0804.90122
6. [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., 1998). Zbl0890.05002
7. [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, Inc., 1998). Zbl0883.00011
8. [8] S.M. Hedetniemi, S.T. Hedetniemi and T.V. Wimer, Linear time resource allocation algorithms for trees, Technical report URI -014, Department of Mathematics, Clemson University (1987). Zbl0643.68093
9. [9] E. Sampathkumar, The global domination number of a graph, J. Math. Phys. Sci. 23 (1989) 377-385. Zbl0729.05045
10. [10] E.R. Scheinerman and D.H. Ullman, Fractional Graph Theory: A Rational Approch to the Theory of Graphs (John Wiley & Sons, New York, 1997). Zbl0891.05003

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.