Résumé
For every$r \in \mathbb{N}$ , let$\theta_r$denote the graph with two vertices and$r$parallel edges. The$\theta_r$ -girth of a graph$G$is the minimum number of edges of a subgraph of$G$that can be contracted to$\theta_r$ . This notion generalizes the usual concept of girth which corresponds to the case$r=2$ . In [Minors in graphs of large girth, Random Structures & Algorithms, 22(2):213--225, 2003], Kühn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is an exponential function of their girth. We extend this result for the case of$\theta_{r}$ -girth and we show that the minimum degree can be replaced by some connectivity measurement. As an application of our results, we prove that, for every fixed$r$ , graphs excluding as a minor the disjoint union of$k$$\theta_{r}$ 's have treewidth$O(k\cdot \log k)$ .