In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph
H, the class of H-graphs, defined as the intersection graphs of connected
subgraphs of some subdivision of H. 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 H-graphs
for different graphs H. In this work we undertake this research topic,
focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and
Zeman showed, for every fixed tree T, a polynomial-time algorithm recognizing
T-graphs. Tucker showed a polynomial time algorithm recognizing K3-graphs
(circular-arc graphs). On the other hand, Chaplick at al. showed that
recognition of H-graphs is NP-hard if H contains two different cycles
sharing an edge.
The main two results of this work narrow the gap between the NP-hard and
P cases of H-graphs recognition. First, we show that recognition of
H-graphs is NP-hard when H contains two different cycles. On the other
hand, we show a polynomial-time algorithm recognizing L-graphs, where L is
a graph containing a cycle and an edge attached to it (L-graphs are called
lollipop graphs). Our work leaves open the recognition problems of M-graphs
for every unicyclic graph M different from a cycle and a lollipop. Other
results of this work, which shed some light on the cases that remain open, are
as follows. Firstly, the recognition of M-graphs, where M is a fixed
unicyclic graph, admits a polynomial time algorithm if we restrict the input to
graphs containing particular holes (hence recognition of M-graphs is probably
most difficult for chordal graphs). Secondly, the recognition of medusa graphs,
which are defined as the union of M-graphs, where M runs over all unicyclic
graphs, is NP-complete