Optimal Locating-Total Dominating Sets in Strips of Height 3

Ville Junnila

Discussiones Mathematicae Graph Theory (2015)

  • Volume: 35, Issue: 3, page 447-462
  • ISSN: 2083-5892

Abstract

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A set C of vertices in a graph G = (V,E) is total dominating in G if all vertices of V are adjacent to a vertex of C. Furthermore, if a total dominating set C in G has the additional property that for any distinct vertices u, v ∈ V C the subsets formed by the vertices of C respectively adjacent to u and v are different, then we say that C is a locating-total dominating set in G. Previously, locating-total dominating sets in strips have been studied by Henning and Jafari Rad (2012). In particular, they have determined the sizes of the smallest locating-total dominating sets in the finite strips of height 2 for all lengths. Moreover, they state as open question the analogous problem for the strips of height 3. In this paper, we answer the proposed question by determining the smallest sizes of locating-total dominating sets in the finite strips of height 3 as well as the smallest density in the infinite strip of height 3.

How to cite

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Ville Junnila. "Optimal Locating-Total Dominating Sets in Strips of Height 3." Discussiones Mathematicae Graph Theory 35.3 (2015): 447-462. <http://eudml.org/doc/271210>.

@article{VilleJunnila2015,
abstract = {A set C of vertices in a graph G = (V,E) is total dominating in G if all vertices of V are adjacent to a vertex of C. Furthermore, if a total dominating set C in G has the additional property that for any distinct vertices u, v ∈ V C the subsets formed by the vertices of C respectively adjacent to u and v are different, then we say that C is a locating-total dominating set in G. Previously, locating-total dominating sets in strips have been studied by Henning and Jafari Rad (2012). In particular, they have determined the sizes of the smallest locating-total dominating sets in the finite strips of height 2 for all lengths. Moreover, they state as open question the analogous problem for the strips of height 3. In this paper, we answer the proposed question by determining the smallest sizes of locating-total dominating sets in the finite strips of height 3 as well as the smallest density in the infinite strip of height 3.},
author = {Ville Junnila},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {locating-total dominating set; domination; square grid; strip.; strip},
language = {eng},
number = {3},
pages = {447-462},
title = {Optimal Locating-Total Dominating Sets in Strips of Height 3},
url = {http://eudml.org/doc/271210},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Ville Junnila
TI - Optimal Locating-Total Dominating Sets in Strips of Height 3
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 3
SP - 447
EP - 462
AB - A set C of vertices in a graph G = (V,E) is total dominating in G if all vertices of V are adjacent to a vertex of C. Furthermore, if a total dominating set C in G has the additional property that for any distinct vertices u, v ∈ V C the subsets formed by the vertices of C respectively adjacent to u and v are different, then we say that C is a locating-total dominating set in G. Previously, locating-total dominating sets in strips have been studied by Henning and Jafari Rad (2012). In particular, they have determined the sizes of the smallest locating-total dominating sets in the finite strips of height 2 for all lengths. Moreover, they state as open question the analogous problem for the strips of height 3. In this paper, we answer the proposed question by determining the smallest sizes of locating-total dominating sets in the finite strips of height 3 as well as the smallest density in the infinite strip of height 3.
LA - eng
KW - locating-total dominating set; domination; square grid; strip.; strip
UR - http://eudml.org/doc/271210
ER -

References

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  7. [7] T.W. Haynes, M.A. Henning and J. Howard, Locating and total dominating sets in trees, Discrete Appl. Math. 154 (2006) 1293-1300. doi:10.1016/j.dam.2006.01.002[Crossref] Zbl1091.05051
  8. [8] M.A. Henning and N. Jafari Rad, Locating-total domination in graphs, Discrete Appl. Math. 160 (2012) 1986-1993. doi:10.1016/j.dam.2012.04.004[Crossref][WoS] 
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