  Neighbourly set of a graph is a subset of edges which either share an end
point or are joined by an edge of that graph. The maximum cardinality
neighbourly set problem is known to be NP-complete for general graphs. Mahdian
(M.Mahdian, On the computational complexity of strong edge coloring, Discrete
Applied Mathematics, 118:239-248, 2002) proved that it is in polynomial time
for quadrilateral-free graphs and proposed an O(n^{11}) algorithm for the same
(along with a note that by a straightforward but lengthy argument it can be
proved to be solvable in O(n^5) running time). In this paper we propose an
O(n^2) time algorithm for finding a maximum cardinality neighbourly set in a
quadrilateral-free graph.
