Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Kings and Kernels in Semicomplete Compositions   Let $T$ be a digraph with vertices $u_1, \dots, u_t$ ($t\ge 2$) and let $H_1,
\dots, H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le
n_i.$ Then the composition $Q=T[H_1, \dots, H_t]$ is a digraph with vertex set
$\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set
$$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le
j_i\le n_i, 1\le q_p\le n_p\}.$$ The composition $Q=T[H_1, \dots, H_t]$ is a
semicomplete composition if $T$ is semicomplete, i.e. there is at least one arc
between every pair of vertices.
  For the topic of kings, we first characterize digraph compositions with a
$k$-king and digraph compositions all of whose vertices are $k$-kings,
respectively. When $T$ is semicomplete, we discuss the existence of 3-kings,
and the adjacency between a 3-king and a non-king in this class of digraphs. We
also study the minimum number of 4-kings in a strong semicomplete composition.
  For the topic of kernels, we first study the existence of a pair of disjoint
quasi-kernels in semicomplete compositions. We then deduce that the problem of
determining whether a strong semicomplete composition has a $k$-kernel is
NP-complete when $k\in \{2,3\}$, and is polynomial time solvable when $k\geq
4$.

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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: supr-con
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The categories of this paper are: econ
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph