Algebraic theory for the clique operator

The complexity of clique graph recognition

“…First articles focused on recognizing clique graphs [20,36], In [4,13], graphs for which the clique graph changes whenever a vertex is removed are considered. Graphs fixed under the operator K or fixed under the iterated clique operator, I<n, for some positive integer n; and the behavior under these operators of parameters such as the number of vertices or diameter were studied in [5,8,9,12,26,30] and more recently in [7,14,[21][22][23]29], For several classes of graphs, the image of the class under the clique operator was characterized [10,18,19,24,34,37]; and, in some cases, also the inverse image of the class [16,28,35], Results of the previous bibliography can be found in the survey [39], Clique graphs have been much studied as intersection graphs and are included in several books [11,25,33],In this paper we are concerned with the time complexity of the problem of recognizing clique graphs, this is the time complexity of the following decision problem.A complete set of a graph G is a subset of vertices inducing a complete subgraph. A clique is a maximal complete set.…”