Niche Number of Linear Hypertrees

Abstract

For a digraph $D$, the niche hypergraph $NH(D)$ of $D$ is the hypergraph having the same set of vertices as $D$ and the set of hyperedges is \begin{align} E(NH(D)) &= \{e \subseteq V(D) : |e| \geq 2~and~there~exists~v \in V(D)~such~that~e = N_{D}^{-}(v)\notag &~~~~~~~or~e = N_{D}^{+}(v)\}.\notag \end{align} A digraph is said to be acyclic if it has no directed cycle as a subdigraph. For a given hypergraph $H$, the niche number $\hat{n}(H)$ is the smallest integer such that $H$ together with $\hat{n}(H)$ isolated vertices is the niche hypergraph of an acyclic digraph. In this paper, we study the niche number of linear hypertrees with maximum degree two. By our result, we can conclude for a special case that if $H$ is a linear hypertree with $\Delta(H) = 2$ and anti-rank three, then $\hat{n}(H) = 0$. We also prove that the maximum degree condition is best possible. Moreover, it was proved that if $H$ is a hypergraph of rank $r$ whose niche number is not infinity, then $\Delta(H) \leq 2r$. In this paper, we give a construction of hypertrees whose niche number is $0$ of prescribed maximum degree from $3$ to $2r$.