A kernelofa digraph G= (V

A) is an independentsubsetof V

say K

such that for any vertex u in VwK there is an arc from u to a vertex v in K. G issaid to be kernel-perfect(KP) if every induced subgraph ofGhas a kernel. Gis said to be kernel-perfect-critical(KPC) ifG has no kernelbutevery proper induced subgraph ofGhas a kernel. The digraph G= (V

A)= C→ n(j1

j2

…

jk) isdifined by: V(G)= ｛0

1

…

n- 1｝

A(G)= ｛uv｜v- u≡ji(m od n)for1ik｝. In this paper

we investigate the digraph G= C→ n(1

±δd

±2d

…

…sd)

denoted by G(n

d

r

s)

where δ= 1 ford> 1 or δ= 0 ford= 1

and n

d

r

sare positiveintegers with (n

d)= r and n= m r. Itis proved thatif(n

d)= r> 2 and s= 1

then G(n

d

r

s) isa KP graph ifand only ifeither (a)m 5 or m = 3

or (b) m = 2 or4 and r isodd; and G(n

d

r

s) isa KPCgraph ifand only ifm = 2 or4 and r is even. Forthecaseofr= 2 and s= 1

som e sufficientconditionsforG(n

d

2

1) to be KP and a conjecture forG(n

d

2

1) to be KPCare also given.