DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 20(2) (2000) 271-280
DOI: 10.7151/dmgt.1126

DOMINATION AND INDEPENDENCE SUBDIVISION NUMBERS OF GRAPHS

Teresa W. Haynes

Department of Mathematics
East Tennessee State University
Johnson City, TN 37614 USA

Sandra M. Hedetniemi

Stephen T. Hedetniemi

Department of Computer Science
Clemson University
Clemson, SC 29634 USA

Abstract

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.

Keywords: domination, independence, subdivision numbers.

2000 Mathematics Subject Classification: 05C69.

References

[1] S. Arumugam, private communication, June 2000.
[2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973).
[3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier, New York, 1977).
[4] G. Chartrand and L. Lesniak, Graphs & Digraphs (Wadsworth and Brooks/Cole, Monterey, CA, third edition, 1996).
[5] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969).
[6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).
[7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., New York, 1998).
[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.
[9] D.B. West, Introduction to Graph Theory (Prentice Hall, New Jersey, 1996).

Received 4 August 2000