Geodesic
Highly symmetric maps on surfaces with boundary
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 100-121

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The regular maps and the arc-transitive maps on surfaces with non-empty boundary are classified. It is shown that it is unrealistic to expect a similar classification of edge-transitive maps on such surfaces. 
@article{ZNSL_2016_446_a5,
     author = {G. A. Jones},
     title = {Highly symmetric maps on surfaces with boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {100--121},
     publisher = {mathdoc},
     volume = {446},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a5/}
}
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G. A. Jones. Highly symmetric maps on surfaces with boundary. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 100-121. http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a5/