  The aim of the dispersion problem is to place a set of $k(\leq n)$ mobile
robots in the nodes of an unknown graph consisting of $n$ nodes such that in
the final configuration each node contains at most one robot, starting from any
arbitrary initial configuration of the robots on the graph. In this work we
propose a variant of the dispersion problem where we start with any number of
robots, and put an additional constraint that no two adjacent nodes contain
robots in the final configuration. We name this problem as
Distance-2-Dispersion (D-2-D). However, even if the number of robots $k$ is
less than $n$, it may not possible for each robot to find a distinct node to
reside, maintaining our added constraint. Specifically, if a maximal
independent set is already formed by the nodes which contain a robot each, then
other robots, if any, who are searching for a node to seat, will not find one.
Hence we allow multiple robots to seat on some nodes only if there is no place
to seat. If $k\geq n$, it is guaranteed that the nodes with robots form a
maximal independent set of the underlying network.
  The graph $G=(V, E)$ has $n$ nodes and $m$ edges, where nodes are anonymous.
It is a port labelled graph, i.e., each node $u$ assigns a distinct port number
to each of its incident edges from a range $[0,\delta-1]$ where $\delta$ is the
degree of the node $u$. The robots have unique ids in the range $[1, L]$, where
$L \ge k$. Co-located robots can communicate among themselves. We provide an
algorithm that solves D-2-D starting from a rooted configuration (i.e.,
initially all the robots are co-located) and terminate after $2\Delta(8m-3n+3)$
synchronous rounds using $O(log \Delta)$ memory per robot without using any
global knowledge of the graph parameters $m$, $n$ and $\Delta$, the maximum
degree of the graph. We also provide $\Omega(m\Delta)$ lower bound on the
number of rounds for the D-2-D problem.
