3 research outputs found
Niche hypergraphs
If D = (V,A) is a digraph, its niche hypergraph Nℋ(D) = (V, ℰ) has the edge set ℰ = {ℯ ⊆ V | |e| ≥ 2 ∧ ∃ v ∈ V : e = ND-(v) ∨ ℯ = ND+(v)}. Niche hypergraphs generalize the well-known niche graphs (see [11]) and are closely related to competition hypergraphs (see [40]) as well as double competition hypergraphs (see [33]). We present several properties of niche hypergraphs of acyclic digraphs
Triangle-free -step competition graphs
Given a digraph and a positive number , the -step competition graph
of is defined to have the same vertex set as and an edge joining two
distinct vertices and if and only if there exist a -directed
walk and a -directed walk both having length for some vertex in
. We call a graph an -step competition graph if it is the -step
competition graph of a digraph. The notion of -step competition graph was
introduced by Cho \emph{et al.} \cite{cho2000m} as one of the variants of
competition graph which was introduced by Cohen \cite{Cohen} while studying
predator-prey concepts in ecological food webs.
In this paper, we first extend a result given by Helleloid
\cite{helleloid2005connected} stating that for all positive integers , the only connected triangle-free -step competition graph on vertices
is the star graph. We show that the result is true for arbitrary positive
integer as long as it is the -step competition graph of a digraph
having a source. We go further to completely characterize the digraphs each of
whose weak components has a source and whose -step competition graphs are
triangle-free for some integer . We also compute the number of
digraphs with a source whose -step competition graphs are connected and
triangle-free for some integer