  The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree
(resp. in a grid) are well studied graph classes. Recently we introduced the
graph classes Edge-Intersecting and Non-Splitting Paths in a Tree ENPT, and in
a Grid (ENPG). It was shown that ENPG contains an infinite hierarchy of
subclasses that are obtained by restricting the number of bends in the paths.
Motivated by this result, in this work we focus on one bend {ENPG} graphs. We
show that one bend ENPG graphs are properly included in two bend ENPG graphs.
We also show that trees and cycles are one bend ENPG graphs, and characterize
the split graphs and co-bipartite graphs that are one bend ENPG. We prove that
the recognition problem of one bend ENPG split graphs is NP-complete even in a
very restricted subfamily of split graphs. Last we provide a linear time
recognition algorithm for one bend ENPG co-bipartite graphs.
