On Some Generalizations of Outerplanar Graphs: Results and Open Problems

Outerplanar graphs have been recently generalized in many directions. Almost all generalizations have been introduced to parameterize the family of planar graphs so that in consequence some of the decision problems which are NP-complete for planar graphs and easy (or trivial) for outerplanar graphs can be solved in polynomial time for every fixed value of a parameter. In this paper, we survey some families of graphs which are known as generalizations of outerplanar graphs: tree-structured graphs (e.g., series-parallel, k-terminal and Halin graphs), W-outerplanar and k-outerplanar. We discuss also some problems which are formulated by slightly modified versions of the statements that characterize outerplanar graphs: the largest face problem and the (independent) face covers in plane graphs. Almost every (sub)section of the paper contains an open problem, a good starting point for further research. Our description of results is rather informal, the reader interested in details is referred to the original works.

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