Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonempty intersection of longest paths in $2K_2$-free graphs   In 1966, Gallai asked whether all longest paths in a connected graph share a
common vertex. Counterexamples indicate that this is not true in general.
However, Gallai's question is positive for certain well-known classes of
connected graphs, such as split graphs, interval graphs, circular arc graphs,
outerplanar graphs, and series-parallel graphs. A graph is $2K_2$-free if it
does not contain two independent edges as an induced subgraph. In this paper,
we show that in nonempty $2K_2$-free graphs, every vertex of maximum degree is
common to all longest paths. Our result implies that all longest paths in a
nonempty $2K_2$-free graph have a nonempty intersection. In particular, it
gives a new proof for the result on split graphs, as split graphs are
$2K_2$-free.

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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn