  Golumbic, Lipshteyn, and Stern defined in 2009 the class of EPG graphs, the
intersection graph class of edge paths on a grid. An EPG graph $G$ is a graph
that admits a representation where its vertices correspond to paths in a grid
$Q$, such that two vertices of $G$ are adjacent if and only if their
corresponding paths in $Q$ have a common edge. If the paths in the
representation have at most $k$ bends, we say that it is a $B_k$-EPG
representation. A collection $C$ of sets satisfies the Helly property when
every sub-collection of $C$ that is pairwise intersecting has at least one
common element. In this paper, we show that given a graph $G$ and an integer
$k$, the problem of determining whether $G$ admits a $B_k$-EPG representation
whose edge-intersections of paths satisfy the Helly property, so-called
Helly-$B_k$-EPG representation, is in NP, for every $k$ bounded by a polynomial
function of $|V(G)|$. Moreover, we show that the problem of recognizing
Helly-$B_1$-EPG graphs is NP-complete, and it remains NP-complete even when
restricted to 2-apex and 3-degenerate graphs.
