  We study the computational complexity of fair division of indivisible items
in an enriched model: there is an underlying graph on the set of items. And we
have to allocate the items (i.e., the vertices of the graph) to a set of agents
in such a way that (a) the allocation is fair (for appropriate notions of
fairness) and (b) each agent receives a bundle of items (i.e., a subset of
vertices) that induces a subgraph with a specific "nice structure." This model
has previously been studied in the literature with the nice structure being a
connected subgraph. In this paper, we propose an alternative for connectivity
in fair division. We introduce what we call compact graphs, and look for fair
allocations in which each agent receives a compact bundle of items. Through
compactness, we try to capture the idea that every agent must receive a set of
"closely related" items. We prove a host of hardness and tractability results
for restricted input settings with respect to fairness concepts such as
proportionality, envy-freeness and maximin share guarantee.
