On the unavoidability of oriented trees

A digraph is {\it $n$-unavoidable} if it is contained in every tournament of order $n$. We first prove that every arborescence of order $n$ with $k$ leaves is $(n+k-1)$-unavoidable. We then prove that every oriented tree of order $n$ ($n\geq 2$) with $k$ leaves is $(\frac{3}{2}n+\frac{3}{2}k -2)$-unavoidable and $(\frac{9}{2}n -\frac{5}{2}k -\frac{9}{2})$-unavoidable, and thus $(\frac{21}{8} n- \frac{47}{16})$-unavoidable. Finally, we prove that every oriented tree of order $n$ with $k$ leaves is $(n+ 144k^2 - 280k + 124)$-unavoidable

On the unavoidability of oriented trees

International audienc

Finding a subdivision of a digraph

International audienceWe consider the following problem for oriented graphs and digraphs: Given a directed graph D, does it contain a subdivision of a prescribed digraph F? We give a number of examples of polynomial instances, several NP-completeness proofs as well as a number of conjectures and open problems