Introductory topics 
 Planar graphs, representation, MST, dual graph, separators
 Surfaces, classification of surfaces, representations of embedded graphs (rotation systems, polygonal schemata, treecotree decomposition),
 Shortest noncontractible and nonseparating cycles on surfaces
 de Verdiere, Chapter 4 (and the necessary prerequisites from chapter 3)
 (as an alternative, or supplement) Erickson Section 8

Advanced topics 
For most topics, the survey by J.Erickson gives a good summary.
 Shortest path homotopic to a given path in polygons

John Hershberger, Jack Snoeyink: Computing minimum length paths of a given homotopy class. Computational Geometry, Volume 4, 1994, pp. 6397 (pdf)
 Survey, Section 4.1
 Minimumlength homotopy basis with a given basepoint

Jeff Erickson, Kim Whittlesey: Greedy optimal homotopy and homology generators. SODA 2005, pp. 10381046 (pdf)

Eric Colin de Verdiere: Shortest Cut Graph of a Surface with Prescribed Vertex Set. ESA 2010, pp.100111 (pdf)
 Survey, Section 3.2
 Computing a canoncical system of loops

Francis Lazarus, Gert Vegter, Michel Pocchiola, Anne Verroust: Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface. SoCG 2001, pp. 8089 (pdf)
 Survey, Section 3.3
 Optimal first homology basis on surfaces

Jeff Erickson, Kim Whittlesey: Greedy optimal homotopy and homology generators. SODA 2005, pp. 10381046 (pdf)
 Tamal Dey, Jian Sun, Yusu Wang: Approximating Loops in a Shortest Homology Basis from Point Data. SoCG 2010, pp.166175 (pdf)
 Survey, Section 6.1
 Optimal first homology basis on arbitrary complexes
 Chao Chen, Daniel Freedman: Measuring and computing natural generators for homology groups. Computational Geometry, Volume 43, 2010, pp. 169181 (pdf)
 Tamal Dey, Jian Sun, Yusu Wang: Approximating Loops in a Shortest Homology Basis from Point Data. SoCG 2010, pp.166175 (pdf)
 Survey, Section 6.2
 Localization over surfaces in Z_2
 Jeff Erickson, Amir Nayyeri: Minimum cuts and shortest nonseparating cycles via homology covers. SODA 2011, pp. 11661176 (pdf)
 Gross, Tucker: Topological Graph Theory. ISBN13: 9780486417417
 Survey, Section 7
