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# Nonempty intersection of longest paths in series–parallel graphs

Publikationstyp

Journal Article

Publikationsdatum

2017

Sprache

English

Institut

TORE-URI

Enthalten in

Volume

340

Issue

3

Start Page

287

End Page

304

Citation

Discrete Mathematics 3 (340): 287-304 (2017)

Publisher DOI

Scopus ID

Publisher

Elsevier

Gallai 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.