Beyond circular-arc graphs – recognizing lollipop graphs and medusa graphs
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In 1992 Biró, 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, Töpfer, Voborník, and Zeman showed, for every fixed tree T, a polynomial-time algorithm recognizing T-graphs. Tucker showed a polynomial time algorithm recognizing K_3-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.
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