Abstract
Apart from their use in excluded minor theorems, induced subgraphs without complete separators play a key role in the theory of simplicial decompositions of graphs [ 1 ], which gives the above problem some weight of its own. If we bound the order of the complete separators under consideration by some fixed natural number k, then, as Křiž and Thomas [ 2 ] observed, the problem has a straightforward positive solution: if a graph G has no complete separator of order < k, then any finite G ⊂ G can be extended to a finite induced subgraph H ′ of G which has no complete separator of order < k either. The purpose of this note is to settle the general case of the problem:
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