### Book chapter

## Hardness and Algorithms for Variants of Line Graphs of Directed Graphs

Given a directed graph *D* = (*V*,*A*) we define its intersection graph *I*(*D*) = (*A*,*E*) to be the graph having *A* as a node-set and two nodes of *I*(*D*) are adjacent if their corresponding arcs share a common node that is the tail of at least one of these arcs. We call them facility location graphs since they arise from the classical uncapacitated facility location problem. In this paper we show that facility location graphs are hard to recognize but they are easy to recognize when the underlying graph is triangle-free. We also determine the complexity of the vertex coloring, the stable set and the facility location problem for triangle-free facility location graphs.