  The minimum number of inputs needed to control a network is frequently used
to quantify its controllability. Control of linear dynamics through a minimum
set of inputs, however, often has prohibitively large energy requirements and
there is an inherent trade-off between minimizing the number of inputs and
control energy. To better understand this trade-off, we study the problem of
identifying a minimum set of input nodes such that controllabililty is ensured
while restricting the length of the longest control chain. The longest control
chain is the maximum distance from input nodes to any network node, and recent
work found that reducing its length significantly reduces control energy. We
map the longest control chain-constraint minimum input problem to finding a
joint maximum matching and minimum dominating set. We show that this graph
combinatorial problem is NP-complete, and we introduce and validate a heuristic
approximation. Applying this algorithm to a collection of real and model
networks, we investigate how network structure affects the minimum number of
inputs, revealing, for example, that for many real networks reducing the
longest control chain requires only few or no additional inputs, only the
rearrangement of the input nodes.
