Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dichotomizing $k$-vertex-critical $H$-free graphs for $H$ of order four   For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical
$(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-free
graphs have at most $2k-1$ vertices. Together with previous research, these
results imply the following characterization where $H$ is a graph of order
four: There is a finite number of $k$-vertex-critical $H$-free graphs for fixed
$k \geq 5$ if and only if $H$ is one of $\overline{K_4}, P_4, P_2 + 2P_1$, or
$P_3 + P_1$. Our results imply the existence of new polynomial-time certifying
algorithms for deciding the $k$-colorability of $(P_2+\ell P_1)$-free graphs
for fixed $k$.

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The categories of this paper are: comp-gas
19
The categories of this paper are: eess
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
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The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin