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Clique Graphs, k-null Ordered Sets

In [48] S. Hazan and V. Neumann-Lara investigate clique graphs: If G=(V,E) is a graph, then its clique graph k(G)   has the maximal cliques of G as vertices and two maximal cliques are connected by an edge iff they intersect.   An ordered set is called k-null iff there is an tex2html_wrap_inline12222 such that the n-th iterated clique graph of the comparability graph is the one point graph. They prove that k-null ordered sets have the fixed point property.
This result provides a new algorithm to approach the fixed point property, namely by computation of clique graphs. The theory in that direction seems to be unexplored and even though the algorithm might be inefficient, the connection is quite surprising and very interesting.