Fractional Domination Game

Abstract

Given a graph $G$, a real-valued function $f: V(G) \rightarrow [0,1]$ is a fractional dominating function if $\sum_{u \in N[v]} f(u) \ge 1$ holds for every vertex $v$ and its closed neighborhood $N[v]$ in $G$. The aim is to minimize the sum $\sum_{v \in V(G)} f(v)$.
 A different approach to graph domination is the domination game, introduced by Brešar et al. [SIAM J. Discrete Math. 24 (2010) 979–991]. It is played on a graph $G$ by two players, namely Dominator and Staller, who take turns choosing a vertex such that at least one previously undominated vertex becomes dominated. The game is over when all vertices are dominated. Dominator wants to finish the game as soon as possible, while Staller wants to delay the end. Assuming that both players play optimally and Dominator starts, the length of the game on $G$ is uniquely determined and is called the game domination number of $G$.
 We introduce and study the fractional version of the domination game, where the moves are ruled by the condition of fractional domination. Here we prove a fundamental property of this new game, namely the fractional version of the so-called Continuation Principle. Moreover, we present lower and upper bounds on the fractional game domination number of paths and cycles. These estimates are tight apart from a small additive constant. We also prove that the game domination number cannot be bounded above by any linear function of the fractional game domination number.

Indicated total domination game

“…The game ends when there are no legal moves, so when the set of played vertices is a dominating set of G. The goal of Dominator is to finish the game with minimum number of moves, while the aim of Staller is to maximize the number of moves. If both players play optimally, then the number of moves played on G is the invariant (see [2,5]) the game domination number γ g (G), which is the number of moves on G if Dominator starts the game. We refer to the domination game when Dominator makes the A c c e p t e d m a n u s c r i p t first move in the game as the D-game (standing for 'Dominator start game').…”

“…We show next that γ tg (T k ) ≥ 5 2 k. Let the total domination D-game be played on T k . We provide a strategy for Staller that forces at least 5 2 k moves to be made in the game. We note that when the game has ended, the vertex w i has been played in order to totally dominate the leaf x i for each i ∈ [k].…”

“…Therefore upon completion of the total domination Dgame, Staller guarantees that at least three moves are played from at least k/2 of the sets {u j , v j , w j , x j }, and from the remainder of the sets {u j , v j , w j , x j } at least two moves are played. Therefore, when the game is complete at least 3 × k 2 + 2 × k 2 = 5 2 k moves were made, implying that γ tg (T k ) ≥ 5 2 k. Finally, let the indicated total domination game be played on T k . We give a strategy for Dominator that ensures at most 2k + 2 vertices are chosen.…”

Optimal strategies in fractional games: vertex cover and domination

“…The important motivation of this approach are the domination game [7] and the total domination game [17], where in fact the transversal game is played on the 'closed neighborhood hypergraph' and on the 'open neighborhood hypergraph' of a graph, respectively. 3 Further variants studied so far include the disjoint domination [14], connected domination [2], and fractional domination [15] games on graphs, and the domination games on hypergraphs [13]. Some of the most recent results can be found in [3,4,8,11,12,16,18,19,20,21,22,23].…”

The domination game played on diameter 2 graphs

“…[8]. For additional topics of interest related to the domination game and its variants see [2,4,6,11,16,17,22,23,25,26].…”