Geodesic
Clique irreducibility of some iterative classes of graphs
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 2, pp. 307-321

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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 
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Aparna Lakshmanan, S.; Vijayakumar, A. Clique irreducibility of some iterative classes of graphs. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 2, pp. 307-321. http://geodesic.mathdoc.fr/item/DMGT_2008_28_2_a6/