Graph grabbing game on totally-weighted graphs

The graph grabbing game is a two-player game on a connected graph with a vertex-weight function. In the game, they alternately remove a non-cut vertex from the graph (i.e., the resulting graph remains connected) and get the weight assigned to the vertex. Each player's aim is to maximize his or her outcome, when all vertices have been taken. In this paper, we consider the graph grabbing game on totally-weighted graphs that are graphs with weight functions from a set of elements in the vertex set and the edge set to non-negative real numbers. In this version, when a player removes a non-cut vertex v, that player gets the weight of v plus the total weight assigned to the edges incident to v. In particular, we give some results of interest for the graph grabbing game on edge-weighted trees, i.e., every vertex has weight zero. Moreover, we consider the game on edge-weighed graphs in the altered rule that each player must keep the connectedness of graphs induced by edges.