# Domination and independence subdivision numbers of graphs

• Volume: 20, Issue: 2, page 271-280
• ISSN: 2083-5892

top

## Abstract

top
The domination subdivision number $s{d}_{\gamma }\left(G\right)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of adjacent vertices in G. We then define the independence subdivision number $s{d}_{\beta }\left(G\right)$ to equal the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the independence number. We show that for any graph G of order n ≥ 2, either $G={K}_{1,m}$ and $s{d}_{\beta }\left(G\right)=m$, or $1\le s{d}_{\beta }\left(G\right)\le 2$. We also characterize the graphs G for which $s{d}_{\beta }\left(G\right)=2$.

## How to cite

top

Teresa W. Haynes, Sandra M. Hedetniemi, and Stephen T. Hedetniemi. "Domination and independence subdivision numbers of graphs." Discussiones Mathematicae Graph Theory 20.2 (2000): 271-280. <http://eudml.org/doc/270298>.

@article{TeresaW2000,
abstract = {The domination subdivision number $sd_γ(G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of adjacent vertices in G. We then define the independence subdivision number $sd_β(G)$ to equal the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the independence number. We show that for any graph G of order n ≥ 2, either $G = K_\{1,m\}$ and $sd_β(G) = m$, or $1 ≤ sd_β(G) ≤ 2$. We also characterize the graphs G for which $sd_β(G) = 2$.},
author = {Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination; independence; subdivision numbers; dominating set; domination number; domination subdivision number; independence subdivision numbers},
language = {eng},
number = {2},
pages = {271-280},
title = {Domination and independence subdivision numbers of graphs},
url = {http://eudml.org/doc/270298},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Teresa W. Haynes
AU - Sandra M. Hedetniemi
AU - Stephen T. Hedetniemi
TI - Domination and independence subdivision numbers of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2000
VL - 20
IS - 2
SP - 271
EP - 280
AB - The domination subdivision number $sd_γ(G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of adjacent vertices in G. We then define the independence subdivision number $sd_β(G)$ to equal the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the independence number. We show that for any graph G of order n ≥ 2, either $G = K_{1,m}$ and $sd_β(G) = m$, or $1 ≤ sd_β(G) ≤ 2$. We also characterize the graphs G for which $sd_β(G) = 2$.
LA - eng
KW - domination; independence; subdivision numbers; dominating set; domination number; domination subdivision number; independence subdivision numbers
UR - http://eudml.org/doc/270298
ER -

## References

top
1. [1] S. Arumugam, private communication, June 2000.
2. [2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973).
3. [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier, New York, 1977). Zbl1226.05083
4. [4] G. Chartrand and L. Lesniak, Graphs & Digraphs (Wadsworth and Brooks/Cole, Monterey, CA, third edition, 1996).
6. [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). Zbl0890.05002
7. [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., New York, 1998). Zbl0883.00011
8. [8] G. Hopkins and W. Staton, Graphs with unique maximum independent sets, Discrete Math. 57 (1985) 245-251, doi: 10.1016/0012-365X(85)90177-3. Zbl0583.05034
9. [9] D.B. West, Introduction to Graph Theory (Prentice Hall, New Jersey, 1996). Zbl0845.05001

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.