Characterization of common-edge sigraph

A sigraph   is a graph GG in which each edge xx carries a value s(x)∈{−1,+1}s(x)∈{−1,+1} called its sign  , denoted specially as S=(G,s)S=(G,s). Given a sigraph SS, a new sigraph CE(S)CE(S), called the common-edge sigraph   of SS is that sigraph whose vertex-set is the set of pairs of adjacent edges in SS and two vertices of CE(S)CE(S) are adjacent if the corresponding pairs of adjacent edges of SS have exactly one edge in common, and the sign of the edge is the sign of the common edge. If all the edges of the sigraph SS carry + sign then SS is actually a graph and the corresponding common-edge sigraph is termed as the common-edge graph. In this paper, we characterize common-edge graph and common-edge sigraph and write an algorithm to obtain a corresponding common-edge root graph and common-edge root sigraph from a given common-edge graph and common-edge sigraph respectively.