# Distance independence in graphs

• Volume: 31, Issue: 2, page 397-409
• ISSN: 2083-5892

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## Abstract

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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 ${\beta }_{D}\left(G\right)$ is the maximum cardinality of a D-independent set. In particular, the independence number $\beta \left(G\right)={\beta }_{1}\left(G\right)$. Along with general results we consider, in particular, the odd-independence number ${\beta }_{ODD}\left(G\right)$ where ODD = 1,3,5,....

## How to cite

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J. Louis Sewell, and Peter J. Slater. "Distance independence in graphs." Discussiones Mathematicae Graph Theory 31.2 (2011): 397-409. <http://eudml.org/doc/270818>.

@article{J2011,
abstract = {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 $β_D(G)$ 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 $β_\{ODD\}(G)$ where ODD = 1,3,5,....},
author = {J. Louis Sewell, Peter J. Slater},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {independence number; distance set},
language = {eng},
number = {2},
pages = {397-409},
title = {Distance independence in graphs},
url = {http://eudml.org/doc/270818},
volume = {31},
year = {2011},
}

TY - JOUR
AU - J. Louis Sewell
AU - Peter J. Slater
TI - Distance independence in graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 2
SP - 397
EP - 409
AB - 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 $β_D(G)$ 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 $β_{ODD}(G)$ where ODD = 1,3,5,....
LA - eng
KW - independence number; distance set
UR - http://eudml.org/doc/270818
ER -

## References

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1. [1] E.J. Cockayne, S.T. Hedetniemi, and D.J. Miller, Properties of hereditary hypergraphs and middle graphs, Canad. Math. Bull. 21 (1978) 461-468, doi: 10.4153/CMB-1978-079-5. Zbl0393.05044
2. [2] T. Gallai, Über extreme Punkt-und Kantenmengen, Ann. Univ. Sci. Budapest, Eotvos Sect. Math. 2 (1959) 133-138.
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4. [4] J.D. McFall and R. Nowakowski, Strong indepedence in graphs, Congr. Numer. 29 (1980) 639-656.
5. [5] J.L. Sewell, Distance Generalizations of Graphical Parameters, (Univ. Alabama in Huntsville, 2011).
6. [6] A. Sinko and P.J. Slater, Generalized graph parametric chains, submitted for publication.
7. [7] A. Sinko and P.J. Slater, R-parametric and R-chromatic problems, submitted for publication.
8. [8] P.J. Slater, Enclaveless sets and MK-systems, J. Res. Nat. Bur. Stan. 82 (1977) 197-202. Zbl0421.05053
9. [9] P.J. Slater, Generalized graph parametric chains, in: Combinatorics, Graph Theory and Algorithms (New Issues Press, Western Michigan University 1999) 787-797.
10. [10] T.W. Haynes, M.A. Henning and P.J. Slater, Strong equality of upper domination and independence in trees, Util. Math. 59 (2001) 111-124. Zbl0980.05038
11. [11] T.W. Haynes, M.A. Henning and P.J. Slater, Strong equality of domination parameters in trees, Discrete Math. 260 (2003) 77-87, doi: 10.1016/S0012-365X(02)00451-X. Zbl1020.05051
12. [12] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, LP-duality, complementarity and generality of graphical subset problems, in: Domination in Graphs Advanced Topics, T.W. Haynes et al. (eds) (Marcel-Dekker, Inc. 1998) 1-30.

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