Discussiones Mathematicae Graph Theory 31(2) (2011) 397-409
doi: 10.7151/dmgt.1554

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Distance Independence in Graphs

J. Louis Sewell

Department of Mathematical Sciences
University of Alabama in Huntsville
Huntsville, AL 35899 USA

Peter J. Slater

Department of Mathematical Sciences
and Computer Sciences Department
University of Alabama in Huntsville
Huntsville, AL 35899 USA


For a set D of positive integers, we define a vertex set S ⊆ V(G) to be D-independent if u, v ∈ S implies the distance d(u, v) ∉ D. The D-independence number βD(G) is the maximum cardinality of a D-independent set. In particular, the independence number β(G) = β{1}(G). Along with general results we consider, in particular, the odd-independence number βODD(G) where ODD = {1,3,5,…}.

Keywords: independence number, distance set

2010 Mathematics Subject Classification: 05C12, 05C38, 05C69, 05C70, 05C76.


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Received 4 January 2010
Revised 6 January 2011
Accepted 10 January 2011