Cycle Base Structures Of Outerplanar Graphs On Surfaces With Small Genera

In this paper, firstly, we mainly study the cycle base structures of 2-connected outerplanar graphs on the plane by topological methods. Furthermore, we find that the minimal cycle base of 2-connected outerplanar graphs on the plane is only determined by the facial cycles. This generalizes the results of Josef Leydold and Peter F.Stadler and other scholars. Secondly, we investigate the cycle base structures of 2-connected outerplanar graphs on the projective plane. Then we show that there is a one-to-one correspondence between minimal cycle base and the shortest noncontractible cycle of embedded graph G when the embedding face-width fω(G) ≥ 2 and edge-width eω(G) ≥ 5. This provides a way to count the number of minimal cycle base via the set of noncontractible cycles. Finally, we investigate the cycle base structures of 2-connected graphs on the torus and the Klein bottle and prove that there is a one-to-one correspondence between the minimal cycle base and two nonhomotopic noncontractible cycles with the shortest total length when fω(G)≥ 2 and eω(G) ≥ m, m = max{l_i | 1 ≤ i<≤f/}(l₁,...,l_f denote the length of all the non-Hamilton facial walks of G). Also, this gives a way to enumerate the minimal cycle base by noncontractible cycles.