Clique‐convergence is undecidable for automatic graphs

Abstract

The clique operator transforms a graph G into its clique graph K ( G ), which is the intersection graph of all the (maximal) cliques of G. Iterated clique graphs are then defined by K n ( G ) = K ( K n − 1 ( G ) ), K 0 ( G ) = G. If there are some n ≠ m such that K n ( G ) ≅ K m ( G ), then we say that G is clique‐convergent. The clique graph operator and iterated clique graphs have been studied extensively, but no characterization for clique‐convergence has been found so far. Automatic graphs are (not necessarily finite) graphs whose vertices and edges can be recognized by finite automata. Automatic graphs (and automatic structures) have strong decidability properties inherited from the finite automata defining them. Here we prove that clique‐convergence is algorithmically undecidable for the class of automatic graphs. Moreover, the problem remains undecidable, if we reduce to the class that contain only quasi‐clique‐Helly and bounded degree graphs. As a consequence, it follows that clique‐convergence for automatic graphs is not first‐order expressible.

On the clique behavior and Hellyness of the complements of regular graphs

“…If this sequence of graphs has a finite number of different graphs up to isomorphism, we say that G is convergent, otherwise, G is said to be divergent. There is a number of criteria in the literature in order to determine which of these behaviors correspond to a given graph G, however it is conjectured that the problem in general is algorithmically unsolvable for finite graphs ( [1]).…”

On the clique behavior of graphs of low degree

“…The problem of determining the behavior of G under iterated applications of the clique operator is one of the main topics in this theory, as the clique graph operator is considered one of the most complex graph operators ( [17]). There are many families of graphs for which criterions have been proved, and in some cases the behavior can even be determined in polynomial time (see for example [7], [8], [11], [9], [10], [13], [14], [6], [4], [1]), however the problem has been found to be undecidable for automatic graphs ( [2]).…”