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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International audienceA set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by inline image, is the minimum cardinality of an independent dominating set. In this article, we show that if inline image is a connected cubic graph of order n that does not have a subgraph isomorphic to K2, 3, then inline image. As a consequence of our main result, we deduce Reed's important result [Combin Probab Comput 5 (1996), 277–295] that if G is a cubic graph of order n, then inline image, where inline image denotes the domination number of G

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The domination game is played on a graph G by two players, named Dominator and Staller. They alternatively select vertices of G such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator's goal is that the game is finished as soon as possible, while Staller wants the game to last as long as possible. It is assumed that both play optimally. Game 1 and Game 2 are variants of the game in which Dominator and Staller has the first move, respectively. The game domination number γg(G), and the Staller-start game domination number γ ′ g (G), is the number of vertices chosen in Game 1 and Game 2, respectively. It is proved that if e ∈ E(G), then |γg(G) − γg(G − e)| ≤ 2 and |γ Possibilities here are again realizable by connected graphs G in almost all the cases, the exceptional values are treated similarly as in the edge-removal case.

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