In 1992 Biró, Hujter and Tuza introduced, for every fixed connected graph , the class of -graphs, defined as the intersection graphs of connected subgraphs of some subdivision of . Such classes of graphs are related to many known graph classes: for example, -graphs coincide with interval graphs, -graphs with circular-arc graphs, the union of -graphs, where ranges over all trees, coincides with chordal graphs. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of -graphs for different graphs . In this work we undertake this research topic, focusing on the recognition problem. Chaplick, Töpfer, Voborník, and Zeman showed an XP-algorithm testing whether a given graph is a -graph, where the parameter is the size of the tree . In particular, for every fixed tree the recognition of -graphs can be solved in polynomial time. Tucker showed a polynomial time algorithm recognizing -graphs (circular-arc graphs). On the other hand, Chaplick et al. showed also that for every fixed graph containing two distinct cycles sharing an edge, the recognition of -graphs is NP-hard. The main two results of this work narrow the gap between the NP-hard and P cases of -graph recognition. First, we show that the recognition of -graphs is NP-hard when contains two distinct cycles. On the other hand, we show a polynomial-time algorithm recognizing -graphs, where is a graph containing a cycle and an edge attached to it (which we call lollipop graphs). Our work leaves open the recognition problems of -graphs for every unicyclic graph different from a cycle and a lollipop.