# Some Toughness Results in Independent Domination Critical Graphs

• Volume: 35, Issue: 4, page 703-713
• ISSN: 2083-5892

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## Abstract

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A subset S of V (G) is an independent dominating set of G if S is independent and each vertex of G is either in S or adjacent to some vertex of S. Let i(G) denote the minimum cardinality of an independent dominating set of G. A graph G is k-i-critical if i(G) = k, but i(G+uv) < k for any pair of non-adjacent vertices u and v of G. In this paper, we establish that if G is a connected 3-i-critical graph and S is a vertex cutset of G with |S| ≥ 3, then [...] improving a result proved by Ao [3], where ω(G−S) denotes the number of components of G−S. We also provide a characteriza- tion of the connected 3-i-critical graphs G attaining the maximum number of ω(G − S) when S is a minimum cutset of size 2 or 3.

## How to cite

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Nawarat Ananchuen, and Watcharaphong Ananchuen. "Some Toughness Results in Independent Domination Critical Graphs." Discussiones Mathematicae Graph Theory 35.4 (2015): 703-713. <http://eudml.org/doc/275986>.

@article{NawaratAnanchuen2015,
abstract = {A subset S of V (G) is an independent dominating set of G if S is independent and each vertex of G is either in S or adjacent to some vertex of S. Let i(G) denote the minimum cardinality of an independent dominating set of G. A graph G is k-i-critical if i(G) = k, but i(G+uv) < k for any pair of non-adjacent vertices u and v of G. In this paper, we establish that if G is a connected 3-i-critical graph and S is a vertex cutset of G with |S| ≥ 3, then [...] improving a result proved by Ao [3], where ω(G−S) denotes the number of components of G−S. We also provide a characteriza- tion of the connected 3-i-critical graphs G attaining the maximum number of ω(G − S) when S is a minimum cutset of size 2 or 3.},
author = {Nawarat Ananchuen, Watcharaphong Ananchuen},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination critical; toughness},
language = {eng},
number = {4},
pages = {703-713},
title = {Some Toughness Results in Independent Domination Critical Graphs},
url = {http://eudml.org/doc/275986},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Nawarat Ananchuen
AU - Watcharaphong Ananchuen
TI - Some Toughness Results in Independent Domination Critical Graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 4
SP - 703
EP - 713
AB - A subset S of V (G) is an independent dominating set of G if S is independent and each vertex of G is either in S or adjacent to some vertex of S. Let i(G) denote the minimum cardinality of an independent dominating set of G. A graph G is k-i-critical if i(G) = k, but i(G+uv) < k for any pair of non-adjacent vertices u and v of G. In this paper, we establish that if G is a connected 3-i-critical graph and S is a vertex cutset of G with |S| ≥ 3, then [...] improving a result proved by Ao [3], where ω(G−S) denotes the number of components of G−S. We also provide a characteriza- tion of the connected 3-i-critical graphs G attaining the maximum number of ω(G − S) when S is a minimum cutset of size 2 or 3.
LA - eng
KW - domination critical; toughness
UR - http://eudml.org/doc/275986
ER -

## References

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1. [1] N. Ananchuen and W. Ananchuen, A characterization of independent domination critical graphs with a cutvertex , J. Combin. Math. Combin. Comput. (to appear). Zbl06587652
2. [2] N. Ananchuen, W. Ananchuen and L. Caccetta, A characterization of connected 3-i-critical graphs of connectivity two, (2014) submitted.
3. [3] S. Ao, Independent Domination Critical Graphs, Master Thesis (University of Victoria, 1994).
4. [4] M. Dehmer, (Ed.), Structural Analysis of Complex Networks (Birkhauser, Bre- ingsville, 2011). Zbl1201.05002
5. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (Eds), Domination in Graphs: Ad- vanced Topics (Marcel Dekker, New York, 1998).
6. [6] D.P. Sumner and P. Blitch, Domination critical graphs, J. Combin. Theory Ser. B 34 (1983) 65-76. doi:10.1016/0095-8956(83)90007-2 [Crossref] Zbl0512.05055

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