Characterization of outerplanar graphs unavoidable with a blue edge

Characterize all outerplanar graphs H for which there exists an integer N(H) at least |H| such that every maximal outerplanar graph on n vertices, for every n at least N(H), contains a copy of H.

Background

The paper completely characterizes many unavoidable pairs in maximal outerplanar graphs, but the case of pairs {H, K_2} reduces to determining which graphs occur as subgraphs of every sufficiently large maximal outerplanar graph. Equivalently, this asks for a characterization of outerplanar graphs whose outerplanar Turán number is eventually smaller than the maximum number of edges in a maximal outerplanar graph.

The problem is introduced as a continuation of the discussion in Remark 2.14, where the authors note that existing outerplanar Turán results provide partial information but do not yield a full characterization. Solving it would complete the unresolved structural part of the {H, K_2} case for maximal outerplanar Ramsey theory.