In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure, is emphasized. Many related topics, including questions on the complexity of related problems, are investigated.

In this paper we introduce a domination game based on the notion of connected domination. Let G = (V, E) be a connected graph of order at least 2. We define a connected domination game on G as follows: The game is played by two players, Dominator and Staller. The players alternate taking turns choosing a vertex of G (Dominator starts). A move of a player by choosing a vertex v is legal, if (1) the vertex v dominates at least one additional vertex that was not dominated by the set of previously chosen vertices and (2) the set of all chosen vertices induces a connected subgraph of G. The game ends when none of the players has a legal move (i.e., G is dominated). The aim of Dominator is to finish as soon as possible, Staller has an opposite aim. Let D be the set of played vertices obtained at the end of the connected domination game (D is a connected dominating set of G). The connected game domination number of G, denoted γcg(G), is the minimum cardinality of D, when both players played optimally on G. We provide an upper bound on γcg(G) in terms of the connected domination number. We also give a tight upper bound on this parameter for the class of 2-trees. Next, we investigate the Cartesian product of a complete graph and a tree, and we give exact values of the connected game domination number for such a product, when the tree is a path or a star. We also consider some variants of the game, in particular, a Staller-start game.

A well-established generalization of graph coloring is the concept of list coloring. In this setting, each vertex v of a graph G is assigned a list L(v) of k colors and the goal is to find a proper coloring c of G with c(v) ∈ L(v). The smallest integer k for which such a coloring c exists for every choice of lists is called the list chromatic number of G and denoted by l (G).We study list colorings of Cartesian products of graphs. We show that unlike in the case of ordinary colorings, the list chromatic

We prove: (1) that ch P (G) − χ P (G) can be arbitrarily large, where ch P (G) and χ P (G) are P-choice and P-chromatic numbers, respectively, (2) the (P, L)-colouring version of Brooks' and Gallai's theorems.