Nonempty intersection of longest paths in series–parallel graphs

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Nonempty intersection of longest paths in series-parallel graphs

In 1966 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 ...

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In 1966, Gallai asked whether every connected graph has a vertex that is common to all its longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs. Another related question was raised in 1995 at the British Combinatorial Conference: Do any three longest paths in a connected graph have a vertex in common? We prove that, in a connected g...

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Longest Paths in Series-Parallel Graphs

In 1966 Gallai asked whether all longest paths in a connected graph intersect in a common vertex. This is not true in general and various counterexamples have been found. However, many well-known classes of graphs have the socalled Gallai property, e.g. trees and outerplanar graphs. Recently, de Rezende et al. proved that 2-trees, which are maximal series-parallel graphs, also have this propert...

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ژورنال

عنوان ژورنال: Discrete Mathematics

سال: 2017

ISSN: 0012-365X

DOI: 10.1016/j.disc.2016.07.023