This paper gives a simple algorithm for solving a class of graphical games where infinite play is possible. A Deterministic Graphical (DG) game is a two person zero sum game played on a directed graph with n > o nodes. Nodes are of two kinds: terminal and continuing. Terminal nodes are those with no successors, and have a payoff to player 1 associated with them. Continuing nodes have at least one successor, and are labelled to indicate which player chooses the successor. Play begins at some specified node, and continues until a terminal node is reached. If no terminal node is ever reached, the payoff is by convention O. The author's main intention in this paper is to describe an algorithm for solving DG games in o(n cubed) steps.