Understanding Fractional Domination I am finding difficulty in understanding the definition of Fractional Domination.
Definition
A function $f:V \rightarrow [0,1]$ is called a fractional domination function if for every vertex $u \in V$, $f(N[u]) \geq 1$. The gractional domination number $\gamma_f(G)$ of a graph $G$ equals the minimum weight of a fractional dominating function $f$ on $G$.
I would really appreciate if you can help me understand this definition with some examples.
 A: A function $f : V \to [0, 1]$ is a fractional domination function if, for any $u \in V$,
$$f(u) + \sum_{(u, v) \in E} f(v) \ge 1. \tag{1}$$
That is, the sum of the function value at a vertex $u$ and all of its neighbours, is at least $1$.
Here's a very simple example: the constant function $1$. This is a fractional domination function on any graph, since $(1)$ involves at least one term, each of which is $1$.
The paper also provides a class of examples that are almost as simple. If a graph is $k$-regular, then the constant function $f(v) = \frac{1}{k+1}$ is a fractional domination function, as the sum in $(1)$ will have $k + 1$ terms ($k$ neighbours, plus one for the vertex itself), each of which contributes $\frac{1}{k + 1}$ to the sum.
Here's another example: take the complete bipartite graph $K_{m,n}$, and assign the part with $m$ vertices each $\frac{1}{m}$, and the part with $n$ vertices each $\frac{1}{n}$. I'll let you do the maths yourself, but the sum in $(1)$ will come out to $1 + \frac{1}{n}$ or $1 + \frac{1}{m}$ depending on which part you're in.

Given such a function $f$, the weight of the function is given by
$$f(V) = \sum_{v \in V} f(v).$$
The fractional domination number is the minimum weight achievable by fractional domination functions.
Example 2.17 from the paper also claims that, in the second example, produces such a minimal weight. If the $k$-regular graph has $n$ vertices, then this minimal weight should be $\frac{n}{k+1}$.
We can prove this with a double-counting argument. Suppose that $g$ is a fraction domination function of a $k$-regular graph. Consider the sum
$$m = \sum_{(u, v) \in E} g(u) = k \sum_{u \in V} g(u),$$
since the graph is $k$-regular. We can also express $m$ by summing over each vertex's neighbourhood in the following way:
$$m = \sum_{v \in V} \left(\sum_{(u, v) \in E} g(u) \right) \ge \sum_{v \in V} (1 - g(v)),$$
since $g$ is a fractional domination function. Thus,
$$k \sum_{u \in V} g(u) \ge \sum_{v \in V} (1 - g(v)) = n - \sum_{u \in V} g(u),$$
and so,
$$\sum_{u \in V} g(u) \ge \frac{n}{k+1}.$$
