Online sum-paintability: The slow-coloring game

We prove V(G)2α(G)+12≤s̊(G)V(G)≤maxV(H)α(H):H⊂G, where α(G) is the independence number, and we study when equality holds in the bounds. We compute s̊(G) for graphs with α(G)=2. Among n-vertex trees, we prove that s̊ is minimized by the star and maximized by the path. We also study s̊(Kr,s).