# Clique irreducibility of some iterative classes of graphs

Aparna Lakshmanan S.; A. Vijayakumar

Discussiones Mathematicae Graph Theory (2008)

- Volume: 28, Issue: 2, page 307-321
- ISSN: 2083-5892

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topAparna Lakshmanan S., and A. Vijayakumar. "Clique irreducibility of some iterative classes of graphs." Discussiones Mathematicae Graph Theory 28.2 (2008): 307-321. <http://eudml.org/doc/270328>.

@article{AparnaLakshmananS2008,

abstract = {In this paper, two notions, the clique irreducibility and clique vertex irreducibility are discussed. A graph G is clique irreducible if every clique in G of size at least two, has an edge which does not lie in any other clique of G and it is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G. It is proved that L(G) is clique irreducible if and only if every triangle in G has a vertex of degree two. The conditions for the iterations of line graph, the Gallai graphs, the anti-Gallai graphs and its iterations to be clique irreducible and clique vertex irreducible are also obtained.},

author = {Aparna Lakshmanan S., A. Vijayakumar},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {line graphs; Gallai graphs; anti-Gallai graphs; clique irreducible graphs; clique vertex irreducible graphs},

language = {eng},

number = {2},

pages = {307-321},

title = {Clique irreducibility of some iterative classes of graphs},

url = {http://eudml.org/doc/270328},

volume = {28},

year = {2008},

}

TY - JOUR

AU - Aparna Lakshmanan S.

AU - A. Vijayakumar

TI - Clique irreducibility of some iterative classes of graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2008

VL - 28

IS - 2

SP - 307

EP - 321

AB - In this paper, two notions, the clique irreducibility and clique vertex irreducibility are discussed. A graph G is clique irreducible if every clique in G of size at least two, has an edge which does not lie in any other clique of G and it is clique vertex irreducible if every clique in G has a vertex which does not lie in any other clique of G. It is proved that L(G) is clique irreducible if and only if every triangle in G has a vertex of degree two. The conditions for the iterations of line graph, the Gallai graphs, the anti-Gallai graphs and its iterations to be clique irreducible and clique vertex irreducible are also obtained.

LA - eng

KW - line graphs; Gallai graphs; anti-Gallai graphs; clique irreducible graphs; clique vertex irreducible graphs

UR - http://eudml.org/doc/270328

ER -

## References

top- [1] Aparna Lakshmanan S., S.B. Rao and A. Vijayakumar, Gallai and anti-Gallai graphs of a graph, Math. Bohem. 132 (2007) 43-54. Zbl1174.05116
- [2] R. Balakrishnan and K. Ranganathan, A Text Book of Graph Theory (Springer, 1999).
- [3] L. Chong-Keang and P. Yee-Hock, On graphs without multicliqual edges, J. Graph Theory 5 (1981) 443-451, doi: 10.1002/jgt.3190050416.
- [4] V.B. Le, Gallai graphs and anti-Gallai graphs, Discrete Math. 159 (1996) 179-189, doi: 10.1016/0012-365X(95)00109-A. Zbl0864.05031
- [5] E. Prisner, Graph Dynamics (Longman, 1995).
- [6] E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220. Zbl0794.05113
- [7] W.D. Wallis and G.H. Zhang, On maximal clique irreducible graphs, J. Combin. Math. Combin. Comput. 8 (1990) 187-193. Zbl0735.05052

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