  We study the classic problem of \emph{fairly} dividing a heterogeneous and
divisible resource -- modeled as a line segment $[0,1]$ and typically called as
a \emph{cake} -- among $n$ agents. This work considers an interesting variant
of the problem where agents are embedded on a graph. The graphical constraint
entails that each agent evaluates her allocated share only against her
neighbors' share. Given a graph, the goal is to efficiently find a
\emph{locally envy-free} allocation where every agent values her share of the
cake to be at least as much as that of any of her neighbors' share.
  The most significant contribution of this work is a bounded protocol that
finds a locally envy-free allocation among $n$ agents on a tree graph using
$n^{O(n)}$ queries under the standard Robertson-Webb (RW) query model. The
query complexity of our proposed protocol, though exponential, significantly
improves the currently best known hyper-exponential query complexity bound of
Aziz and Mackenzie [AM16] for complete graphs. In particular, we also show that
if the underlying tree graph has a depth of at most two, one can find a locally
envy-free allocation with $O(n^4 \log n)$ RW queries. This is the first and the
only known locally envy-free cake cutting protocol with polynomial query
complexity for a non-trivial graph structure.
  Interestingly, our discrete protocols are simple and easy to understand, as
opposed to highly involved protocol of [AM16]. This simplicity can be
attributed to their recursive nature and the use of a single agent as a
designated \emph{cutter}. We believe that these results will help us improve
our algorithmic understanding of the arguably challenging problem of envy-free
cake-cutting by uncovering the bottlenecks in its query complexity and its
relation to the underlying graph structures.
