Stochastic dynamic matching problems have recently gained attention in the\nstochastic-modeling community due to their diverse applications, such as\nsupply-chain management and kidney exchange programs. In this paper, we study a\nmatching problem where items of different classes arrive according to\nindependent Poisson processes. Unmatched items are stored in a queue, and\ncompatibility between items is represented by a simple graph, where items can\nbe matched if their classes are connected.We analyze matching policies in terms\nof stability, delay, and long-term matching rate optimization. Our approach\nrelies on the conservation equation, which ensures a balance between arrivals\nand departures in any stable system. Our main contributions are as follows.We\nestablish a link between the existence of stable policies, the dimensionality\nof the solution set of the conservation equation, and the compatibility graph's\nstructure.We describe the convex polytope formed by non-negative solutions to\nthe conservation equation, and we design policies that can achieve or closely\napproximate the vertices of this polytope.Lastly, we discuss potential\nextensions of our results beyond the main assumptions of this paper.\n