Maximal outerplane graphs of extremal diameter

We explore the maximal outerplane graphs (MOP-graphs) with extremal values of diameter. For arbitrary MOP-graphs we determine the lower and upper bounds of the diameter. For lattice MOP-graphs (i. e., graphs embedded into the lattice of equilateral triangles without "holes" and intersections) we prove that the upper bound of the diameter matches that of the arbitrary MOP-graphs; a preliminary lower bound of the diameter is determined. For the lower and upper bounds of the diameter of arbitrary and lattice MOP-graphs we determine the extremal graphs where these bounds are reached. Extremal graphs with maximal diameter are the same for both arbitrary and lattice MOP-graphs. The obtained results can be used for classification of images represented by MOP-graphs, and for classification of isomers of conjugated polyene hydrocarbons.