  A simple-triangle graph (also known as a PI graph) is the intersection graph
of a family of triangles defined by a point on a horizontal line and an
interval on another horizontal line. The recognition problem for
simple-triangle graphs was a longstanding open problem, and recently a
polynomial-time algorithm has been given [G. B. Mertzios, The Recognition of
Simple-Triangle Graphs and of Linear-Interval Orders is Polynomial, SIAM J.
Discrete Math., 29(3):1150--1185, 2015]. Along with the approach of this paper,
we show a simpler recognition algorithm for simple-triangle graphs. To do this,
we provide a polynomial-time algorithm to solve the following problem: Given a
bipartite graph $G$ and a set $F$ of edges of $G$, find a 2-chain subgraph
cover of $G$ such that one of two chain subgraphs has no edges in $F$.
