Chen, GuantaoGuantaoChenEhrenmüller, JuliaJuliaEhrenmüllerFernandes, Cristina G.Cristina G.FernandesHeise, Carl GeorgCarl GeorgHeiseShan, SonglingSonglingShanYang, PingPingYangYates, Amy N.Amy N.Yates2019-12-182019-12-182017Discrete Mathematics 3 (340): 287-304 (2017)http://hdl.handle.net/11420/4169Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series–parallel if it does not contain K4 as a minor. Series–parallel graphs are also known as partial 2-trees, which are arbitrary subgraphs of 2-trees. We present two independent proofs that every connected series–parallel graph has a vertex that is common to all of its longest paths. Since 2-trees are maximal series–parallel graphs, and outerplanar graphs are also series–parallel, our result captures these two classes in one proof and strengthens them to a larger class of graphs. We also describe how one such vertex can be found in linear time.en1872-681X20173287304ElsevierMathematikJournal Article10.1016/j.disc.2016.07.023Other