48 research outputs found
Some recent results on niche graphs
AbstractIn an earlier paper entitled “Niche graphs” written by Cable, Jones, Lundgren and Seager, niche graphs were introduced and examples were provided of graphs which have niche number 0, 1, 2, and ∞. However, no examples were found of a niche graph having finite niche number 3 or larger. We still have had no success in our efforts to find such a graph. Nevertheless we have gotten some interesting results. For example, we show in this paper that if there is such a graph, then there must be one which is connected. We also show that the niche number of a graph which has a finite niche number is ≤23|V(G)|. In addition we determine the niche number of all “wheel” graphs
The niche graphs of interval orders
The niche graph of a digraph is the (simple undirected) graph which has
the same vertex set as and has an edge between two distinct vertices
and if and only if or , where (resp. ) is the set
of out-neighbors (resp. in-neighbors) of in . A digraph is
called a semiorder (or a unit interval order) if there exist a real-valued
function on the set and a positive real number such that if and only if .
A digraph is called an interval order if there exists an assignment
of a closed real interval to each vertex such that if and only if .
S. -R. Kim and F. S. Roberts characterized the competition graphs of
semiorders and interval orders in 2002, and Y. Sano characterized the
competition-common enemy graphs of semiorders and interval orders in 2010. In
this note, we give characterizations of the niche graphs of semiorders and
interval orders.Comment: 7 page
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
The (1,2)-Step Competition Graph of a Tournament
The competition graph of a digraph, introduced by Cohen in 1968, has been extensively studied. More recently, in 2000, Cho, Kim, and Nam defined the m-step competition graph. In this paper, we offer another generalization of the competition graph. We define the (1,2)-step competition graph of a digraph D, denoted C1,2(D), as the graph on V(D) where {x,y}∈E(C1,2(D)) if and only if there exists a vertex z≠x,y, such that either dD−y(x,z)=1 and dD−x(y,z)≤2 or dD−x(y,z)=1 and dD−y(x,z)≤2. In this paper, we characterize the (1,2)-step competition graphs of tournaments and extend our results to the (i,k)-step competition graph of a tournament