Discussiones Mathematicae Graph Theory 28(2) (2008) 307-321
doi: 10.7151/dmgt.1407

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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

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Received 9 October 2007
Revised 18 March 2008
Accepted 18 March 2008