A subset of vertices of a graph G is a general position set if no triple of vertices from the set lie on a common shortest path in G. In this paper we introduce the general position polynomial as $$\sum _{i \ge 0} a_i x^i$$∑ i ≥ 0 a i x i , where $$a_i$$a i is the number of distinct general position sets of G with cardinality i. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs K(n, 2), with unimodal general position polynomials are presented.

The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P₃ with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with K_k for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.