Discussiones
Mathematicae Graph Theory 20(2) (2000) 271-280
DOI: https://doi.org/10.7151/dmgt.1126
DOMINATION AND INDEPENDENCE SUBDIVISION NUMBERS OF GRAPHS
Teresa W. Haynes Department of Mathematics |
Sandra M. Hedetniemi Stephen T. Hedetniemi Department of Computer Science |
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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Received 4 August 2000
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