  Given a simple undirected graph $G$ and a positive integer $k$, the
$k$-forcing number of $G$, denoted $F_k(G)$, is the minimum number of vertices
that need to be initially colored so that all vertices eventually become
colored during the discrete dynamical process described by the following rule.
Starting from an initial set of colored vertices and stopping when all vertices
are colored: if a colored vertex has at most $k$ non-colored neighbors, then
each of its non-colored neighbors becomes colored. When $k=1$, this is
equivalent to the zero forcing number, usually denoted with $Z(G)$, a recently
introduced invariant that gives an upper bound on the maximum nullity of a
graph. In this paper, we give several upper bounds on the $k$-forcing number.
Notable among these, we show that if $G$ is a graph with order $n \ge 2$ and
maximum degree $\Delta \ge k$, then $F_k(G) \le \frac{(\Delta-k+1)n}{\Delta - k
+ 1 +\min{\{\delta,k\}}}$. This simplifies to, for the zero forcing number case
of $k=1$, $Z(G)=F_1(G) \le \frac{\Delta n}{\Delta+1}$. Moreover, when $\Delta
\ge 2$ and the graph is $k$-connected, we prove that $F_k(G) \leq
\frac{(\Delta-2)n+2}{\Delta+k-2}$, which is an improvement when $k\leq 2$, and
specializes to, for the zero forcing number case, $Z(G)= F_1(G) \le
\frac{(\Delta -2)n+2}{\Delta -1}$. These results resolve a problem posed by
Meyer about regular bipartite circulant graphs. Finally, we present a
relationship between the $k$-forcing number and the connected $k$-domination
number. As a corollary, we find that the sum of the zero forcing number and
connected domination number is at most the order for connected graphs.
