The domination game played on a graph G consists of two players, Dominator and Staller who alternate taking turns choosing a vertex from G such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible and Staller wishes to delay the process as much as possible. The game domination number γ g (G) is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number γ g (G) when Staller starts the game. An imagination strategy is developed as a general tool for proving results on the domination game. We show that for any graph G, γ(G) ≤ γ g (G) ≤ 2γ(G) − 1, and that all possible values can be realized. It is proved that for any graph G, γ g (G) − 1 ≤ γ g (G) ≤ γ g (G) + 2, and that most of the possibilities for mutual values of γ g (G) and γ g (G) can be realized. A connection with Vizing's conjecture is established, and a lower bound on the * The paper was initiated during the visit of the third author to the IMFM and completed during the visit of the first two authors at Furman University supported in part by the Slovenia-USA bilateral grant BI-US/08-10-004 and the Wylie Enrichment Fund of Furman University.† Supported by the Ministry of Science of Slovenia under the grants P1-0297 and J1-2043. The author is also with the Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana.game domination number of an arbitrary Cartesian product is proved. Several problems and conjectures are also stated.

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. ᭧

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A vertex in a graph totally dominates another vertex if they are adjacent. A sequence of vertices in a graph G is called a total dominating sequence if every vertex v in the sequence totally dominates at least one vertex that was not totally dominated by any vertex that precedes v in the sequence, and at the end all vertices of G are totally dominated. While the length of a shortest such sequence is the total domination number of G, in this paper we investigate total dominating sequences of maximum length, which we call the Grundy total domination number, γ t gr (G), of G. We provide a characterization of the graphs G for which γ t gr (G) = |V (G)| and of those for which γ t gr (G) = 2. We show that if T is a nontrivial tree of order n with no vertex with two or more leaf-neighbors, then γ t gr (T ) ≥ 2 3 (n + 1), and characterize the extremal trees. We also prove that for k ≥ 3, if G is a connected k-regular graph of order n different from K k,k , then γ t gr (G) ≥ (n + ⌈ k 2 ⌉ − 2)/(k − 1) if G is not bipartite and γ t gr (G) ≥ (n + 2⌈ k 2 ⌉ − 4)/(k − 1) if G is bipartite. The Grundy total domination number is proven to be bounded from above by two times the Grundy domination number, while the former invariant can be arbitrarily smaller than the latter. Finally, a natural connection with edge covering sequences in hypergraphs is established, which in particular yields the NP-completeness of the decision version of the Grundy total domination number.

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Assume we have a set of k colors and to each vertex of a graph G we assign an arbitrary subset of these colors. If we require that each vertex to which an empty set is assigned has in its neighborhood all k colors, then this is called the k-rainbow dominating function of a graph G. The corresponding invariant γ rk (G), which is the minimum sum of numbers of assigned colors over all vertices of G, is called the k-rainbow domination number of G. In this paper we connect this new concept to usual domination in (products of) graphs, and present its application to paired-domination of Cartesian products of graphs. Finally, we present a linear algorithm for determining a minimum 2-rainbow dominating set of a tree.

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A set S of vertices of a graph G is a geodetic set if every vertex of G lies in at least one interval between the vertices of S. The size of a minimum geodetic set in G is the geodetic number of G. Upper bounds for the geodetic number of Cartesian product graphs are proved and for several classes exact values are obtained. It is proved that many metrically defined sets in Cartesian products have product structure and that the contour set of a Cartesian product is geodetic if and only if their projections are geodetic sets in factors.