對一連通有向圖D中，令S為頂點集V(D)的一子集。若對S內任意兩頂點x,y，每條最短x-y有向路徑及最短y-x有向路徑上的頂點皆落在S內，則稱S為D的一個凸子集。且D的最大凸子集且為真子集的集合大小即為D的凸集數，記為con(D)。
令S_{C}(K_{n})={con(D)|D為一n頂點競賽圖}且S_{SC}(K_{n})={con(D)|D為一n頂點強連通競賽圖}，我們得到S_{C}(K_{3})={1,2}及當n大於等於4時，S_{C}(K_{n})={1,3,4,...,n-1}。我們也得到S_{SC}(K_{3})={1}及當n大於等於4時，S_{SC}(K_{n})={1,3,4,...,n-2}。
我們也找到對任意的整數n、m、k，當n大於等於5、k介於3到n-2之間且m介於n+1到n(n-1)/2之間時，存在一強連通有向圖D的頂點數為n、邊數為m，且凸集數為k。

Given a connected oriented graph D, we say that a subset S of V(D) is convex in D if, for every pair of vertices x, y in S, the vertex set of every x-y geodesic (x-y shortest dipath) and y-x geodesic in D is contained in S. The convexity number con (D) of a nontrivial connected oriented graph D is the maximum cardinality of a proper convex set of D.
Let S_{C}(K_{n})={con(D)|D is an orientation of K_{n}} and S_{SC}(K_{n})={con(D)|D is a strong orientation of K_{n}}. We show that S_{C}(K_{3})={1,2} and S_{C}(K_{n})={1,3,4,...,n-1} if n >= 4. We also have that S_{SC}(K_{3})={1} and S_{SC}(K_{n})={1,3,4,...,n-2} if n >= 4 .
We also show that every triple n, m, k of integers with n >= 5, 3 <= k <= n-2, and n+1 <= m <= n(n-1)/2, there exists a strong connected oriented graph D of order n with |E(D)|=m and con (D)=k.

1 Introduction
2 Previous results
3 The main results
3-1 Strong convexity spectra of complete graphs
3-2 Constructing oriented graphs with fixed order, size, and convexity number
4 References

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