Fractional Domination Game

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.

Domination and Fractional Domination in Digraphs

In this paper, we investigate the relation between the (fractional) domination number of a digraph $G$ and the independence number of its underlying graph, denoted by $\alpha(G)$. More precisely, we prove that every digraph $G$ on $n$ vertices has fractional domination number at most $2\alpha(G)$ and domination number at most $2\alpha(G) \cdot \log{n}$. Both bounds are sharp.