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The p-competition graph Cp(D) of a digraph D = (V,A) is a graph with V (Cp(D)) = V (D), where an edge between distinct vertices x and y if and only if there exist p distinct vertices v1, v2, ..., vp 2 V such that x → vi, y → vi are arcs of the digraph D for each i = 1, 2, ..., p. In this paper, we obtain that a path with order n ≥ 6 is a p-competition graph if and only if n ≥ p + 3. We also show that for a p-competition graph G with no isolated vertices, G+Kn is a p+n-competition graph. And we obtain that a wheel Wn is a p-competition graph if and only if n ≥ p + 3. [ABSTRACT FROM AUTHOR]