Soft Eulerian Graphs

Abstract

Let  = (V, E) be a simple graph and A⊆ ???? ( ) be any nonempty set of parameters. Let ???? be an arbitrary relation from A to V where (F, A) and (K, A) are soft sets over V and E respectively. H(a) = (F(a), K(a)) is an induced subgraph of  for all a ∈ A. ( , F, K, A) =

{H(a)/ A}     is called as the soft graph of  corresponding to the parameter set A and the relation ????. It is said to be a T1- soft graph of  only if H(a) is connected A. Otherwise, it is called a T12-soft graph of  . Every T1-soft graph is also a T12-soft graph of  and not the converse. The geodetic set of the soft graph ( , F, K, A) introduced by K Palani et al. [7] is defined as the union of geodetic sets of the induced sub graphs H(a) where a ∈ A. A geodetic set of a T1 or T12- soft graph of  of minimum cardinality is said to be a minimum geodetic set of ( , F, K, A). The geodetic number of the soft graph ( , F, K, A) is the cardinality of a minimum geodetic set of ( , F, K, A). A connected graph G is called an Euler graph, if there is a closed trail which includes every edge of the graph. This paper extends the eulerian concept to soft graphs and develops the concept of soft eulerian graphs