We investigate random processes for generating task-dependency graphs of order $n$ with $m$ edges and a specified number of initial vertices and terminal vertices. In order to do so, we consider two random processes for generating task-dependency graphs that can be combined to accomplish this task. In the $(x, y)$ edge-removal process, we start with a maximally connected task-dependency graph and remove edges uniformly at random as long as they do not cause the number of initial vertices to exceed $x$ or the number of terminal vertices to exceed $y$. In the $(x, y)$ edge-addition process, we start with an empty task-dependency graph and add edges uniformly at random as long as they do not cause the number of initial vertices to be less than $x$ or the number of terminal vertices to be less than $y$. In the $(x, y)$ edge-addition process, we halt if there are exactly $x$ initial vertices and $y$ terminal vertices. For both processes, we determine the values of $x$ and $y$ for which the resulting task-dependency graph is guaranteed to have exactly $x$ initial vertices and $y$ terminal vertices, and we also find the extremal values for the number of edges in the resulting task-dependency graphs as a function of $x$, $y$, and the number of vertices. Furthermore, we asymptotically bound the expected number of edges in the resulting task-dependency graphs. Finally, we define a random process using only edge-addition and edge-removal, and we show that with high probability this random process generates an $(x, y)$ task-dependency graph of order $n$ with $m$ edges.

翻译：本文研究了用于生成具有$n$阶、$m$条边以及指定初始顶点和终止顶点数量的任务依赖图的随机过程。为此，我们考虑了两种可组合实现该目标的随机过程。在$(x, y)$边删除过程中，从最大连通的任务依赖图出发，只要删除的边不会导致初始顶点数超过$x$或终止顶点数超过$y$，则均匀随机地删除边。在$(x, y)$边添加过程中，从空任务依赖图出发，只要添加的边不会导致初始顶点数少于$x$或终止顶点数少于$y$，则均匀随机地添加边。在$(x, y)$边添加过程中，当恰好存在$x$个初始顶点和$y$个终止顶点时终止。对于这两个过程，我们确定了保证结果任务依赖图恰好具有$x$个初始顶点和$y$个终止顶点的$x$和$y$取值，同时找到了结果任务依赖图中边数的极值关于$x$、$y$和顶点数的函数关系。此外，我们渐近地给出了结果任务依赖图中边数的期望上界。最后，我们定义了一个仅使用边添加和边删除的随机过程，并证明该随机过程以高概率生成一个具有$n$阶、$m$条边的$(x, y)$任务依赖图。