  A connected partition is a partition of the vertices of a graph into sets
that induce connected subgraphs. Such partitions naturally occur in many
application areas such as road networks, and image processing. We consider
Balanced Connected Partitions (BCP), where the two classical objectives for BCP
are to maximize the weight of the smallest, or minimize the weight of the
largest component. We study BCP on c-claw-free graphs, the class of graphs that
do not have $K_{1,c}$ as an induced subgraph, and present efficient
(c-1)-approximation algorithms for both objectives. In particular, due to the
(3-)claw-freeness of line graphs, this also implies a 2-approximations for the
edge-partition version of BCP in general graphs.
  In the 1970s Gy\H{o}ri and Lov\'{a}sz showed for natural numbers
$w_1,\dots,w_k$ where $\sum_i w_i$ is the vertex size, that if $G$ is
k-connected, then there exist a connected k-partition with part sizes
$w_1,\dots,w_k$. However, to this day no polynomial algorithm to compute such
partitions exists for k>4. Towards finding such a partition $T_1,\dots, T_k$,
we show how to efficiently compute connected partitions that at least
approximately meet the target weights, subject to the mild assumption that each
$w_i$ is greater than the weight of the heaviest vertex. In particular, we give
a 3-approximation for both the lower and the upper bounded version i.e. we
guarantee that each $T_i$ has weight at least $\frac{w_i}{3}$ or that each
$T_i$ has weight most $3w_i$, respectively. Also, we present a both-side
bounded version that produces a connected partition where each $T_i$ has size
at least $\frac{w_i}{3}$ and at most $\max(\{r,3\}) w_i$, where $r \geq 1$ is
the ratio between the largest and smallest value in $w_1, \dots, w_k$. In
particular for the balanced version, i.e.~$w_1=w_2=, \dots,=w_k$, this gives a
partition with $\frac{1}{3}w_i \leq w(T_i) \leq 3w_i$.
