DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 25(3) (2005) 251-259
DOI: 10.7151/dmgt.1278

DOMINATION AND LEAF DENSITY IN GRAPHS

Anders Sune Pedersen

Department of Mathematics, Aalborg University
Fredrik Bajers Vej 7G, DK 9220 Aalborg, Denmark
e-mail: asp@math.auc.dk

Abstract

The domination number γ(G) of a graph G is the minimum cardinality of a subset D of V(G) with the property that each vertex of V(G)−D is adjacent to at least one vertex of D. For a graph G with n vertices we define ε(G) to be the number of leaves in G minus the number of stems in G, and we define the leaf density ζ(G) to equal ε(G)/n. We prove that for any graph G with no isolated vertex, γ(G) ≤ n(1− ζ(G))/2 and we characterize the extremal graphs for this bound. Similar results are obtained for the total domination number and the partition domination number.

Keywords: bounds; domination number; leaves; partioned domination; total domination number.

2000 Mathematics Subject Classification: Primary 05C69, 05C35; Secondary 05C75.

References

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Received 19 May 2003
Revised 1 October 2003