The 34 -Game Total Domination Conjecture posed by Henning, Klavžar and Rall [Combinatorica, to appear] states that if G is a graph on n vertices in which every component contains at least three vertices, then γtg(G) ≤ 34n, where γtg(G) denotes the game total domination number of G. Motivated by this conjecture, we raise the problem to a higher level by introducing a transversal game in hypergra...

Two players, Dominator and Staller, alternate choosing vertices of a graph G, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of the Dominator is to finish the game as soon as possible, while the aim of the Staller is just the opposite. The game domination number γg(G) is the number of vertices chosen when Dominator starts the game and both pla...

The domination game is played on an arbitrary graph G by two players, Dominator and Staller. It is known that verifying whether the game domination number of a graph is bounded by a given integer k is PSPACE-complete. On the other hand, it is showed in this paper that the problem can be solved for a graph G in O(∆(G) · |V (G)|k) time. In the special case when k = 3 and the graph G considered ha...

Two players, Dominator and Staller, alternate choosing vertices of a graph G, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of the Dominator is to finish the game as soon as possible, while the aim of the Staller is just the opposite. The game domination number g(G) is the number of vertices chosen when Dominator starts the game and both play...

Game theory is one of the possible ways to study information warfare with mathematical models. This paper presents four example games which illustrate the different requirements for an effective playing strategy in information warfare. These games study, how a bold playing strategy can lead to domination, how a mixed playing strategy can reduce domination, how it can be useful to play a dominat...

This thesis deals with the following three independent problems. Pósa proved that if G is an n-vertex graph in which any two nonadjacent vertices have degree sum at least n + k, then G has a spanning cycle containing any specified family of disjoint paths with a total of k edges. We consider the analogous problem for a bipartite graph G with n vertices and parts of equal size. Let F be a subgra...

In the domination game on a graph G, two players called Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of G. Dominator aims to minimize the size of the resulting dominating set, while Staller aims to maximize it. When both players play optimally, the si...

We present a generalization of the so-called σ-game, introduced by Sutner [9], a combinatorial game played on a graph, with relations to cellular automata, as well as odd domination in graphs. A configuration on a graph is an assignment of values in {0, . . . , p− 1} (where p is an arbitrary positive integer) to all the vertices of G. One may think of a vertex v of G as a button the player can ...