Abstract: In practice, wireless networks has the property that
the signal strength attenuates with respect to the distance from the
base station, it could be better if the nodes at two hop away are
considered for better quality of service. In this paper, we propose a
procedure to identify delay preserving substructures for a given
wireless ad-hoc network using a new graph operation G 2 – E (G) =
G* (Edge difference of square graph of a given graph and the
original graph). This operation helps to analyze some induced
substructures, which preserve delay in communication among them.
This operation G* on a given graph will induce a graph, in which 1-
hop neighbors of any node are at 2-hop distance in the original
network. In this paper, we also identify some delay preserving
substructures in G*, which are (i) set of all nodes, which are mutually
at 2-hop distance in G that will form a clique in G*, (ii) set of nodes
which forms an odd cycle C2k+1 in G, will form an odd cycle in G*
and the set of nodes which form a even cycle C2k in G that will form
two disjoint companion cycles ( of same parity odd/even) of length k
in G*, (iii) every path of length 2k+1 or 2k in G will induce two
disjoint paths of length k in G*, and (iv) set of nodes in G*, which
induces a maximal connected sub graph with radius 1 (which
identifies a substructure with radius equal 2 and diameter at most 4 in
G). The above delay preserving sub structures will behave as good
clusters in the original network.