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.
In this paper a concept Q-Ramsey Class of graphs is introduced, where Q is a class of bipartite graphs. It is a generalization of wellknown concept of Ramsey Class of graphs. Some Q-Ramsey Classes of graphs are presented (Theorem 1 and 2). We proved that T 2 , the class of all outerplanar graphs, is not D 1-Ramsey Class (Theorem 3). This results leads us to the concept of acyclic reducible bounds for a hereditary property P. For T 2 we found two bounds (Theorem 4). An improvement, in some sense, of that in Theorem 5 is given.
For a given graph G and a sequence P 1 , P 2 ,. .. , P n of additive hereditary classes of graphs we define an acyclic (P 1 , P 2 ,. .. , P n)colouring of G as a partition (V 1 , V 2 ,. .. , V n) of the set V (G) of vertices which satisfies the following two conditions: 1. G[V i ] ∈ P i for i = 1,. .. , n, 2. for every pair i, j of distinct colours the subgraph induced in G by the set of edges uv such that u ∈ V i and v ∈ V j is acyclic. A class R = P 1 P 2 • • • P n is defined as the set of the graphs having an acyclic (P 1 , P 2 ,. .. , P n)-colouring. If P ⊆ R, then we say that R is an acyclic reducible bound for P. In this paper we present acyclic reducible bounds for the class of outerplanar graphs.
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