Affiliation:
1. School of Mathematical Science, Shanxi University , 030006 , Taiyuan , P. R. China
2. Department of Applied Mathematics, Taiyuan University of Science and Technology , 030024 , Taiyuan , P. R. China
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.