A lower bound for the irredundance number of trees

Michael Poschen; Lutz Volkmann

Discussiones Mathematicae Graph Theory (2006)

  • Volume: 26, Issue: 2, page 209-215
  • ISSN: 2083-5892

Abstract

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Let ir(G) and γ(G) be the irredundance number and domination number of a graph G, respectively. The number of vertices and leaves of a graph G are denoted by n(G) and n₁(G). If T is a tree, then Lemańska [4] presented in 2004 the sharp lower bound γ(T) ≥ (n(T) + 2 - n₁(T))/3. In this paper we prove ir(T) ≥ (n(T) + 2 - n₁(T))/3. for an arbitrary tree T. Since γ(T) ≥ ir(T) is always valid, this inequality is an extension and improvement of Lemańska's result.

How to cite

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Michael Poschen, and Lutz Volkmann. "A lower bound for the irredundance number of trees." Discussiones Mathematicae Graph Theory 26.2 (2006): 209-215. <http://eudml.org/doc/270471>.

@article{MichaelPoschen2006,
abstract = { Let ir(G) and γ(G) be the irredundance number and domination number of a graph G, respectively. The number of vertices and leaves of a graph G are denoted by n(G) and n₁(G). If T is a tree, then Lemańska [4] presented in 2004 the sharp lower bound γ(T) ≥ (n(T) + 2 - n₁(T))/3. In this paper we prove ir(T) ≥ (n(T) + 2 - n₁(T))/3. for an arbitrary tree T. Since γ(T) ≥ ir(T) is always valid, this inequality is an extension and improvement of Lemańska's result. },
author = {Michael Poschen, Lutz Volkmann},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {irredundance; tree; domination},
language = {eng},
number = {2},
pages = {209-215},
title = {A lower bound for the irredundance number of trees},
url = {http://eudml.org/doc/270471},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Michael Poschen
AU - Lutz Volkmann
TI - A lower bound for the irredundance number of trees
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 2
SP - 209
EP - 215
AB - Let ir(G) and γ(G) be the irredundance number and domination number of a graph G, respectively. The number of vertices and leaves of a graph G are denoted by n(G) and n₁(G). If T is a tree, then Lemańska [4] presented in 2004 the sharp lower bound γ(T) ≥ (n(T) + 2 - n₁(T))/3. In this paper we prove ir(T) ≥ (n(T) + 2 - n₁(T))/3. for an arbitrary tree T. Since γ(T) ≥ ir(T) is always valid, this inequality is an extension and improvement of Lemańska's result.
LA - eng
KW - irredundance; tree; domination
UR - http://eudml.org/doc/270471
ER -

References

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  1. [1] E.J. Cockayne, Irredundance, secure domination, and maximum degree in trees, unpublished manuscript (2004). Zbl1233.05143
  2. [2] E.J. Cockayne, P.H.P. Grobler, S.T. Hedetniemi and A.A. McRae, What makes an irredundant set maximal? J. Combin. Math. Combin. Comput. 25 (1997) 213-224. Zbl0907.05032
  3. [3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). Zbl0890.05002
  4. [4] M. Lemańska, Lower bound on the domination number of a tree, Discuss. Math. Graph Theory 24 (2004) 165-169, doi: 10.7151/dmgt.1222. Zbl1063.05035

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