Abstract
The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as many turns as possible. The game ends when all vertices are dominated. The game domination number, denoted by γ g ( G ) (respectively γ g ′ ( G ) ), is the total number of turns when both players play optimally and when Dominator (respectively Staller) starts the game. In this paper, we study a version of this game where the set of chosen vertices is always independent. This version turns out to be another game known as the competition-independence game. The competition-independence game is played on a graph by two players, Diminisher and Sweller. They take turns in constructing maximal independent set M, where Diminisher tries to minimize | M | and Sweller tries to maximize | M | . Note that, actually, it is the domination game in which the set of played vertices is independent. The competition-independence number, denoted by I d ( G ) (respectively I s ( G ) ) is the optimal size of the final independent set in the competition-independence game if Diminisher (respectively Sweller) starts the game. In this paper, we check whether some well-known results in the domination game hold for the competition-independence game. We compare the competition-independence numbers to the game domination numbers. Moreover, we provide a family of graphs such that many parameters are equal. Finally, we present a realization result on the competition-independence numbers.
Highlights
A dominating set of a graph G is a set S of vertices of G such that every vertex in G is an element in S or is adjacent to an element in S
We check whether some well-known results in the domination game hold for the competition-independence game
We start this section by showing that the Continuation Principle for the domination game does not hold for the competition-independence game
Summary
A dominating set of a graph G is a set S of vertices of G such that every vertex in G is an element in S or is adjacent to an element in S. In each round of the game played by Dominator and Staller, there are at most three new dominated vertices. By similar arguments as in case 1, after Dominator plays ui , there are (12n + 2) − 2(3n + 1) − 2 = 6n − 2 undominated vertices and so the number of moves in this case is at most 4n + 2. Staller plays his turn at the vertex u1 At this point, the vertices x j and u j are dominated for 1 ≤ j ≤ 3n + 1, and yi and zi are dominated.
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