Maximum Cardinality Neighbourly Sets in Quadrilateral Free Graphs
  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. 
Neethi K.S. and Sanjeev Saxena
cs.DS math.CO