# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## 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

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