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A graph is a $k$-leaf power of a tree $T$ if its vertices are leaves of $T$and two vertices are adjacent in $T$ if and only if their distance in $T$ is atmost $k$. Then $T$ is a $k$-leaf root of $G$. This notion was introduced byNishimura, Ragde, and Thilikos [2002] motivated by the search for underlyingphylogenetic trees. We study here an extension of the $k$-leaf power graphrecognition problem. This extension is motivated by a new biological questionfor the evaluation of the latteral gene transfer on a population of viruses. Weallow the host graph to slightly differs from a tree and allow some cycles. Infact we study phylogenetic galled networks in which cycles are pairwise vertexdisjoint. We show some structural results and propose polynomial algorithms forthe cases $k=3$ and $k=4$. As a consequence, squares of galled networks canalso be recognized in polynomial time.

Structure and Recognition of 3,4-leaf Powers of Galled Phylogenetic Networks in Polynomial Time