A characterization of locating-total domination edge critical graphs

Mostafa Blidia; Widad Dali

Discussiones Mathematicae Graph Theory (2011)

  • Volume: 31, Issue: 1, page 197-202
  • ISSN: 2083-5892

Abstract

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For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γₜ(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V(G)∖D, N G ( u ) ∩ D ≠ N G ( v ) ∩ D . The locating-total domination number γ L t ( G ) is the minimum cardinality of a locating-total dominating set of G. A graph G is said to be a locating-total domination edge removal critical graph, or just a γ L t + -ER-critical graph, if γ L t ( G - e ) > γ L t ( G ) for all e non-pendant edge of E. The purpose of this paper is to characterize the class of γ L t + -ER-critical graphs.

How to cite

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Mostafa Blidia, and Widad Dali. "A characterization of locating-total domination edge critical graphs." Discussiones Mathematicae Graph Theory 31.1 (2011): 197-202. <http://eudml.org/doc/271018>.

@article{MostafaBlidia2011,
abstract = {For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γₜ(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V(G)∖D, $N_G(u) ∩ D ≠ N_G(v) ∩ D$. The locating-total domination number $γ_L^t(G)$ is the minimum cardinality of a locating-total dominating set of G. A graph G is said to be a locating-total domination edge removal critical graph, or just a $γ_L^\{t+\}$-ER-critical graph, if $γ_L^t(G-e) > γ_L^t(G)$ for all e non-pendant edge of E. The purpose of this paper is to characterize the class of $γ_L^\{t+\}$-ER-critical graphs.},
author = {Mostafa Blidia, Widad Dali},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {locating-domination; critical graph},
language = {eng},
number = {1},
pages = {197-202},
title = {A characterization of locating-total domination edge critical graphs},
url = {http://eudml.org/doc/271018},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Mostafa Blidia
AU - Widad Dali
TI - A characterization of locating-total domination edge critical graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2011
VL - 31
IS - 1
SP - 197
EP - 202
AB - For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γₜ(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V(G)∖D, $N_G(u) ∩ D ≠ N_G(v) ∩ D$. The locating-total domination number $γ_L^t(G)$ is the minimum cardinality of a locating-total dominating set of G. A graph G is said to be a locating-total domination edge removal critical graph, or just a $γ_L^{t+}$-ER-critical graph, if $γ_L^t(G-e) > γ_L^t(G)$ for all e non-pendant edge of E. The purpose of this paper is to characterize the class of $γ_L^{t+}$-ER-critical graphs.
LA - eng
KW - locating-domination; critical graph
UR - http://eudml.org/doc/271018
ER -

References

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  1. [1] M. Blidia and W. Dali, A characterization of a locating-domination edge critical graphs, Australasian J. Combin. 44 (2009) 297-300. Zbl1194.05113
  2. [2] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). Zbl0890.05002
  3. [3] D.P. Sumner and P. Blitch, Domination critical graphs, J. Combin. Theory (B) 34 (1983) 65-76, doi: 10.1016/0095-8956(83)90007-2. Zbl0512.05055
  4. [4] 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. Zbl1091.05051

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