Abstract
Abstract In 2011, Factor and Merz [Discrete Appl. Math. 159 (2011), 100–103] defined the ( 1 , 2 ) \left(1,2) -step competition graph of a digraph. Given a digraph D = ( V , A ) D=\left(V,A) , the ( 1 , 2 ) \left(1,2) -step competition graph of D, denoted C 1 , 2 ( D ) {C}_{1,2}\left(D) , is a graph on V ( D ) V\left(D) , where x y ∈ E ( C 1 , 2 ( D ) ) xy\in E\left({C}_{1,2}\left(D)) if and only if there exists a vertex z ≠ x , y z\ne x,y such that either d D − y ( x , z ) = 1 {d}_{D-y}\left(x,z)=1 and d D − x ( y , z ) ≤ 2 {d}_{D-x}(y,z)\le 2 or d D − x ( y , z ) = 1 {d}_{D-x}(y,z)=1 and d D − y ( x , z ) ≤ 2 {d}_{D-y}\left(x,z)\le 2 . They also characterized the (1, 2)-step competition graphs of tournaments and extended some results to the ( i , j ) \left(i,j) -step competition graphs of tournaments. In this paper, the definition of the (1, 2)-step competition graph of a digraph is generalized to a hypertournament and the (1, 2)-step competition graph of a k-hypertournament is characterized. Also, the results are extended to ( i , j ) \left(i,j) -step competition graphs of k-hypertournaments.
Highlights
Given two integers n and k, n ≥ k > 1, a k-hypertournament T on n vertices is a pair (V, A), where V is a set of vertices, ∣V∣ = n and A is a set of k-tuples of vertices, called arcs, so that for any k-subset S of V, A contains exactly one of the k ! k-tuples whose entries belong to S
A k-hypertournament T is said to be transitive if its vertices are labeled v1, v2,..., vn in such an order so that i < j if and only if vi precedes vj in each arc containing vi and vj
We study the (1, 2)step competition graph of a k-hypertournament and extend Theorems 1.1–1.3 to k-hypertournaments
Summary
The (i, j)-step competition graph of a k-hypertournament T with i ≥ 1 and j ≥ 1, denoted Ci,j(T), is a graph on V(T), where xy ∈ E(Ci,j(T)) if and only if there exist a vertex z ≠ x, y and an (x, z)-path P and a (y, z)-path Q satisfying the following: (a) y ∉ V(P), x ∉ V(Q); (b) l(P) ≤ i and l(Q) ≤ j, or l(Q) ≤ i and l(P) ≤ j; (c) P and Q are arc-disjoint. If xz satisfies (e), i.e. AT∗{x, z} contains exactly an arc a, there exists a vertex w such that w ∈ N+(x). Let G on n vertices be the (1, 2)-step competition graph of some strong k-hypertournament T, where 3 ≤ k ≤ n − 1. The complement Gc of G does not contain K1,3
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