Domination and independence subdivision numbers of graphs
Teresa W. Haynes ; Sandra M. Hedetniemi ; Stephen T. Hedetniemi
Discussiones Mathematicae Graph Theory, Tome 20 (2000), p. 271-280 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

The domination subdivision number sdγ(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 sdβ(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=K1,m and sdβ(G)=m, or 1≤sdβ(G)≤2. We also characterize the graphs G for which sdβ(G)=2.

Publié le : 2000-01-01
EUDML-ID : urn:eudml:doc:270298 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1126,
     author = {Teresa W. Haynes and Sandra M. Hedetniemi and Stephen T. Hedetniemi},
     title = {Domination and independence subdivision numbers of graphs},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {20},
     year = {2000},
     pages = {271-280},
     zbl = {0984.05066},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1126}
}

Teresa W. Haynes; Sandra M. Hedetniemi; Stephen T. Hedetniemi. Domination and independence subdivision numbers of graphs. Discussiones Mathematicae Graph Theory, Tome 20 (2000) pp. 271-280. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1126/

[000] [1] S. Arumugam, private communication, June 2000.

[001] [2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973).

[002] [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier, New York, 1977). | Zbl 1226.05083

[003] [4] G. Chartrand and L. Lesniak, Graphs & Digraphs (Wadsworth and Brooks/Cole, Monterey, CA, third edition, 1996).

[004] [5] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969).

[005] [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). | Zbl 0890.05002

[006] [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., New York, 1998). | Zbl 0883.00011

[007] [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. | Zbl 0583.05034

[008] [9] D.B. West, Introduction to Graph Theory (Prentice Hall, New Jersey, 1996). | Zbl 0845.05001