Clique divergent clockwork graphs and partial orders

Contractibility and the clique graph operator

“…By adapting the example in Section 3, we obtain in Section 4 a comparability graph which is contractible and K-divergent. This gives a new answer to a problem in [8] and [21], and also settles a question that remained unanswered in [14]. In our last two sections we consider the remaining question of whether K-null graphs are always contractible.…”

“…Rival In [8] and Schröder's book [21, p. 160], the question was raised as to whether posets with the FPP do exist with non K-null comparability graph. Two such posets were given in [14], and for both of them the comparability graph is K-divergent and has the same Betti numbers as the circle. In fact, it was remarked in [14] that "we do not know an example of a non K-null graph with the same Betti numbers as the disk".…”

“…Or take the K-null graphs F n , H i n which were studied by Bornstein and Szwarcfiter in [4]: they are Whitney triangulations of the disk, thus contractible. In fact, it is conjectured in [14] that every Whitney triangulation of the disk is K-null, and this has been proved in [13] for the particular case in which each interior vertex has degree at least six.…”

Equivariant collapses and the homotopy type of iterated clique graphs

“…Clockwork graphs were introduced in [8]. We will use the definition from [9]. A cyclically segmented graph G is a graph with a fixed partition (G 0 , G 1 , .…”

“…Since n g (K(G)) = n g (G), n c (K(G)) ≤ n c (G) and the order of K(G) is o(K(G)) = o(G) + n g (G) − n c (G), it follows that G is clique divergent if n g (G) > n c (G). See [8,9] for more details.…”