## CLIQUE IRREDUCIBILITY OF SOME ITERATIVE CLASSES OF GRAPHS

Aparna Lakshmanan S. and A. Vijayakumar

Department of Mathematics
Cochin University of Science and Technology
Cochin-682 022, India
 e-mail: aparna@cusat.ac.in e-mail: vijay@cusat.ac.in

## 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.

Keywords: line graphs, Gallai graphs, anti-Gallai graphs, clique irreducible graphs, clique vertex irreducible graphs.

2000 Mathematics Subject Classification: 05C99.

## References

 [1] Aparna Lakshmanan S., S.B. Rao and A. Vijayakumar, Gallai and anti-Gallai graphs of a graph, Math. Bohem. 132 (2007) 43-54. [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. [5] E. Prisner, Graph Dynamics (Longman, 1995). [6] E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220. [7] W.D. Wallis and G.H. Zhang, On maximal clique irreducible graphs, J. Combin. Math. Combin. Comput. 8 (1990) 187-193.