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- W2288956423 abstract "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], Kuhn 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 $O(kcdot log k)$." @default.
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- W2288956423 date "2015-10-11" @default.
- W2288956423 modified "2023-09-27" @default.
- W2288956423 title "Minors in graphs of large ${theta}_r$-girth" @default.
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