Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $K_4$ as a minor. Series-parallel graphs are also known as
partial 2-trees, which are arbitrary subgraphs of 2-trees. We present a proof
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 this vertex can be found in linear time.

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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys