On 2−simple graphoidal cover of a graph

A 2–graphoidal cover(2–GC) of G is a set ψ of open paths in G such that every path in ψ contains atleast two vertices and every edge is in exactly one path in ψ and every vertex is an internal vertex of at most two paths. The minimum cardinality of 2–GC of G is called 2–Graphoidal covering number of G and is denoted by η2. A 2–simple graphoidal covering (2–SGC) of a graph G is a 2–GC ψ of G such that any two paths in ψ have atmost one vertex in common. The minimum cardinality of 2–SGC ψ of G is known as 2–simple graphoidal covering number of G and is denoted by η2s. This paper discusses the 2–SGC on trees, unicyclic graphs and standard graphs.