Greedy approaches to Gold Grabbing in split graph subclasses

The Gold Grabbing Game is a combinatorial game on vertex-weighted graphs in which two players alternately remove vertices, keeping the graph connected and aiming to maximize the total collected weight. The literature focuses on strategies that guarantee victory; optimality, that is, the maximization of the gain, remains less explored. We investigate the greedy strategy. We show that it is not optimal for general split graphs, but that for the complete split graphs CS(1,n) and CS(2,n) — the latter known as book graphs — the sequence it produces realizes the game value val(G). As a consequence, the first player does not lose when the number of vertices is even.