\begin{document}


\title{\bf Kings and Kernels in Semicomplete Compositions}
\author{Yuefang Sun\\

School of Mathematics and Statistics, Ningbo University,\\Zhejiang 315211, P. R. China, \\yuefangsun2013@163.com}


\date{}
\maketitle

\begin{abstract}
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. Let $k\geq 2$ be an integer, a $k$-king in a digraph $D$ is a vertex which can reach every other vertex by a directed path of length at most $k$ and a non-king is a vertex which is not a 3-king.
A subset $K$ is $k$-independent if for every pair of vertices $x,y \in K$, we have $d(x, y), d(y, x)\geq k$; it is called $\ell$-absorbent if for every $x\in V(D)\setminus K$ there exists $y\in K$ such that $d(x, y)\leq \ell$. A $k$-kernel of $D$ is a $k$-independent and $(k-1)$-absorbent subset of $V(D)$. A kernel is a 2-kernel. A set $K\subseteq V(D)$ is call a quasi-kernel of $D$ if it is independent, and for every vertex $x\in V(D)\setminus K$, there exists $y\in K$ such that $d(x, y)\leq 2$.

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 $k\geq 4$.
\vspace{0.3cm}\\
{\bf Keywords:} king; kernel; quasi-kernel; digraph composition; semicomplete composition.
\vspace{0.3cm}\\ {\bf AMS subject
classification (2020)}: 05C12, 05C20, 05C69, 05C76, 05C85.

\end{abstract}

\section{Introduction}
We refer the readers to \cite{Bang-Jensen-Gutin, Bondy} for graph-theoretical notation and terminology not given here. All digraphs considered in this paper have no parallel arcs
or loops, and paths and cycles are always assumed to be directed, unless otherwise stated. We use $[n]$ to denote the set of all natural numbers from 1 to $n$.

Let $D$ be a digraph. If there is an arc from a vertex $x$ to a vertex $y$ in $D$, then we say that $x$ {\em dominates} $y$ and denote it by $x\rightarrow y$. If there is no arc from $x$ to $y$ we shall use the notation $x\not\rightarrow y$. If $A$ and $B$ are two subdigraphs of $D$ and every vertex of $A$ dominates each vertex of $B$, then we say that $A$ {\em dominates} $B$ and denote it by $A\rightarrow B$.
We shall use $A\Rightarrow B$ to denote that $A$ dominates $B$ and there is no arcs from $B$ to $A$. A {\em strong component} of a digraph $D$ is a maximal subset $S\subseteq V(D)$ such that $d_D(x,y)$ is finite for every pair of distinct vertices $x, y\in S$, where $d_D(x, y)$ denotes the length of a shortest path from $x$ to $y$ (if there is no such path $d_D(x, y)$ is infinite). An {\em initial strong component} is one which has no arcs coming in from any other strong components.
We call a vertex $x$ in a digraph a {\em sink} (resp. {\em source}) if $d^+(x)=0$ (resp. $d^-(x)=0$).

We call a digraph $D$ {\em vertex-pancyclic} if for every $x\in V(D)$ and every integer $k\in \{3, 4, \dots, n\}$, there exists a $k$-cycle through $x$ in $D$.
A digraph $D$ is {\em semicomplete} if for every pair $x,y$ of distinct vertices of $D$, there is at least one arc between $x$ and $y.$ In particular, a {\em tournament} is a semicomplete digraph without 2-cycle. A digraph is {\em semicomplete multipartite} if it is obtained from a complete multipartite graph by replacing every edge by an arc or a pair of opposite arcs. A {\em multipartite~tournament} is a semicomplete multipartite digraph with no 2-cycle.
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 {\em 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 {\em semicomplete composition} if $T$ is semicomplete. If $Q=T[H_1, \dots, H_t]$ and none of the digraphs $H_1, \dots, H_t$ has an arc, then $Q$ is an {\em extension} of $T$. For any set $\Phi$ of digraphs, $\Phi^{ext}$ denotes the (infinite) set of all extensions of digraphs in $\Phi$, which are called {\em extended $\Phi$-digraphs}. By definition, the class of extended semicomplete digraphs is a subclass of semicomplete compositions. 
A digraph $D$ is {\em quasi-transitive}, if for any triple $x,y,z$ of distinct vertices of $D$, if $xy$ and $yz$ are arcs of $D$
then either $xz$ or $zx$ or both are arcs of $D.$
Bang-Jensen and Huang gave a complete characterization of quasi-transitive digraphs by using digraph compositions as shown below.

\begin{thm}\label{intro01}\cite{Bang-Jensen-Huang1}
Let $D$ be a quasi-transitive digraph. Then the following assertions hold:
\begin{description}
\item[(a)] If $D$ is not strong, then there exists a transitive oriented
graph $T$ with vertices $\{u_i\mid i\in [t]\}$ and strong quasi-transitive digraphs $H_1, H_2, \dots, H_t$ such that $D = T[H_1, H_2, \dots, H_t]$, where $H_i$ is substituted for $u_i$, $i\in [t]$.
\item[(b)] If $D$ is strong, then there exists a strong semicomplete
digraph $S$ with vertices $\{v_j\mid j\in [s]\}$ and quasi-transitive digraphs $Q_1, Q_2, \dots, Q_s$ such that $Q_j$ is either a vertex or is non-strong and $D = S[Q_1, Q_2, \dots, Q_s]$, where $Q_j$ is substituted for $v_j$, $j\in [s]$.
\end{description}
\end{thm}

Digraph compositions generalize some families of digraphs, including quasi-transitive digraphs (by Theorem~\ref{intro01}), extended semicomplete digraphs and lexicographic product digraphs (when $H_i$ is the same digraph $H$ for every $i\in [t]$, $Q$ is the lexicographic product of $T$ and $H$, see, e.g., \cite{Hammack}). In particular, strong semicomplete compositions generalize strong quasi-transitive digraphs (by Theorem~\ref{intro01}). To see that strong semicomplete compositions form a significant generalization of strong quasi-transitive digraphs, observe that the Hamiltonian cycle problem is polynomial-time solvable for quasi-transitive digraphs \cite{Gutin4}, but NP-complete for strong semicomplete compositions (see, e.g., \cite{Bang-Jensen-Gutin-Yeo}).
While digraph composition has been used since 1990s to study quasi-transitive digraphs and their generalizations, see, e.g., \cite{Bang-Jensen-Huang1, GSHC}, the study of digraph decompositions in their own right was initiated only recently by Sun, Gutin and Ai in \cite{Sun-Gutin-Ai}, where some results on the existence of arc-disjoint strong spanning subdigraphs were given. After that, Bang-Jensen, Gutin and Yeo \cite{Bang-Jensen-Gutin-Yeo} characterized all strong semicomplete compositions with a pair of arc-disjoint strong spanning subdigraphs. Recently, Gutin and Sun \cite{Gutin-Sun} studied the existence of a pair (it is called ``good pair'') of arc-disjoint in- and out-branchings rooted at the same vertex in this class of digraphs. Especially, they characterized semicomplete compositions with a good pair rooted at the same vertex, which generalizes the corresponding characterization by Bang-Jensen and Huang \cite{Bang-Jensen-Huang1} for quasi-transitive digraphs. Sun \cite{Sun} studied the structural properties of semicomplete compositions, including results on connectivity, paths, cycles, strong spanning subdigraphs and acyclic spanning subdigraphs. These results show that this class of digraphs shares some nice properties of other classes of digraphs, like quasi-transitive digraphs.

A {\em $k$-king} in a digraph $D$ is a vertex which can reach every other vertex by a directed path of length at most $k$. A $k$-king $u$ is {\em strict} if there exists a vertex $v$ such that $d_D(u,v)=k$. A 2-king is called a {\em king}, and a {\em non-king} is a vertex which is not a 3-king. The study of kings in digraphs began with Landau \cite{Landau} and the term king was introduced by Maurer \cite{Maurer}. The concept of $k$-king was first introduced in \cite{Bang-Jensen-Huang2} for the purpose of studying quasi-transitive digraphs. After spending some years dormant, this subject has received a lot of attention lately. Many of the nice existing results for kings in quasi-transitive digraphs admit natural generalizations to $k$-quasi-transitive digraphs \cite{Bang-Jensen-Gutin, GSHC}. 
Since digraph compositions generalize quasi-transitive digraphs, it is natural to study $k$-kings in this class of digraphs.

Let $D=(V(D), A(D))$ be a digraph. A set $K\subseteq V(D)$ is called a {\em kernel} of $D$ if it is an independent set such that every vertex
in $V(D)\setminus K$ dominates some vertex in $K$. The concept of a kernel was introduced by von Neumann and Morgenstern while studying cooperative games \cite{Neumann-Morgenstern}. After that, kernels have been studied in many areas, such as list coloring, game theory and perfect graphs \cite{Boros-Gurvich}, mathematical logic \cite{Bezem-Grabmayer-Walicki} and complexity theory \cite{Walicki-Dyrkolbotn}.
There are many generalizations of this concept. A set $Q\subseteq V(D)$ is called a {\em quasi-kernel} of $D$ if it is independent, and for every vertex $x\in V(D)\setminus K$, there exists $y\in K$ such that $d(x, y)\leq 2$. A subset $K$ is {\em $k$-independent} if for every pair of vertices $x,y \in K$, we have $d(x, y), d(y, x)\geq k$; it is called {\em $\ell$-absorbent} if for every $x\in V(D)\setminus K$ there exists $y\in K$ such that $d(x, y)\leq \ell$. A {\em $k$-kernel} of $D$ is a $k$-independent and $(k-1)$-absorbent subset of $V(D)$. Clearly, a 2-kernel is exactly a kernel. The problem {\sc $k$-Kernel} is determining whether a given digraph has a $k$-kernel.

In this paper, we continue the research of digraph compositions and focus on the topics of kings and kernels in semicomplete compositions. In Section~\ref{sec:king}, kings are discussed. We first give characterizations of all digraph compositions with a $k$-king and digraph compositions all of whose vertices are $k$-kings, respectively, where $k\geq 2$ is an integer (Theorem~\ref{thma}). When $T$ is semicomplete, we first give a sufficient condition to guarantee that $Q$ has at least two 3-kings. In particular,  every strong semicomplete composition has at least two 3-kings (Theorem~\ref{thmd}). The adjacency between 3-kings and non-kings in a strong semicomplete composition is also studied (Theorem~\ref{thmc}). For a semicomplete composition $Q$, if there exists a semicomplete composition $Q'$ which contains $Q$ as an induced subdigraph such that the set of all 3-kings of $Q'$ is precisely $V(Q)$, then we say $Q$ can be {\em established}. Similar problems for tournaments and quasi-transitive digraphs have been studied in \cite{Bang-Jensen-Huang2, Huang-Li, Reid}. In this paper, we give a sufficient condition under which a strong semicomplete composition can be established (Theorem~\ref{thme}). In the end of this section, we discuss the minimum number of 4-kings in a strong semicomplete composition (Theorem~\ref{thmf}).

The topic of kernels are discussed in Section~\ref{sec:Kernel}. 
Gutin et al. \cite{Gutin-Koh-Tay-Yeo} characterized digraphs with exactly one or two quasi-kernels. In particular, if a digraph has precisely two quasi-kernels then these two quasi-kernels are actually disjoint. This raises the question of which digraphs contain a pair of disjoint quasi-kernels. We deduce that when $T$ has no sink, each semicomplete composition contains a pair of disjoint quasi-kernels. In particular, every strong semicomplete composition contains a pair of disjoint quasi-kernels (Theorem~\ref{thmb1}). The problem {\sc $2$-Kernel}, also called {\sc Kernel}, is NP-Complete \cite{Chvatal}, even restricted to planar digraphs with $\Delta\leq 3$ and $\Delta^+, \Delta^-\leq 2$ \cite{Fraenkel}, or digraphs with 3-colourable underlying graph \cite{Hell-Hernandez-Cruz}. However, when restricted to quasi-transitive digraphs, the above problem can be solved in polynomial time \cite{Hell-Hernandez-Cruz}. For the {\sc $3$-Kernel} problem, Hell and Hern\'{a}ndez-Cruz \cite{Hell-Hernandez-Cruz} proved that it is NP-complete even when restricted to cyclically 3-partite digraphs of circumference 6, but can be solved in polynomial time restricted to semicomplete multipartite digraphs or 3-quasi-transitive digraphs. In the end of Section~\ref{sec:Kernel}, we study the complexity of {\sc $k$-Kernel} for any integer $k\geq 2$ and deduce that this problem restricted to strong semicomplete compositions is NP-complete when $k\in \{2,3\}$ and is polynomial-time solvable when $k\geq 4$ (Theorem~\ref{thmd2}).


\section{Kings}\label{sec:king}

\subsection{$T$ is arbitrary}

By definition, we directly have the following observations.
\begin{obs}\label{obs3}\cite{Sun}
Let $Q=T[H_1, \dots, H_t]$ be a digraph composition, $Q$ is strong if and only if $T$ is strong.
\end{obs}


\begin{obs}\label{obs2}
Let $k\geq 2$ be an integer. If $H$ is a spanning subdigraph of $D$, then every $k$-king (if exists) of $H$ is also a $k$-king of $G$.
\end{obs}

\begin{obs}\label{obs1}
Let $k\geq 2$ be an integer. If $Q=T[H_1, \dots, H_t]$ has a $k$-king $v\in V(H_i)$ for some $i \in [t]$, then $u_i$ is a $k$-king of $T$.
\end{obs}

For any integer $k\geq 2$, we now characterize all digraph compositions with a $k$-king and digraph compositions all of whose vertices are $k$-kings, respectively, in the following theorem which will be used in our sequel argument.

\begin{thm}\label{thma}
Let $k\geq 2$ be an integer, and $Q=T[H_1, \dots, H_t]$ be a digraph composition. Then the following assertions hold:
\begin{description}
\item[(a)] $Q$ has a $k$-king if and only if $T$ has a $k$-king $u_{i}$ for some $i\in [t]$ and at least one of the following possibilities holds: $(i)$~$H_{i}$ has a $k$-king; $(ii)$~$|V(H_{i})|\geq 2$ and $u_{i}$ belongs to a cycle of $T$ with length at most $k$.
\item[(b)] All vertices of $Q$ are $k$-kings if and only if for each $i\in [t]$, $u_{i}$ is a $k$-king of $T$ and at least one of the following possibilities holds: $(i)$~all vertices of $H_{i}$ are $k$-kings of $H_i$;   $(ii)$~$|V(H_{i})|\geq 2$ and $u_{i}$ belongs to a cycle of $T$ with length at most $k$.
\end{description}

\end{thm}
\begin{pf}

\noindent{\bf Part (a)} The proof of {\em sufficiency} is not hard, so we just prove the {\em necessity}. If $Q=T[H_1, \dots, H_t]$ has a $k$-king $v$ which belongs to some $V(H_i)$, then $u_i$ is a $k$-king of $T$ by Observation \ref{obs1}. If $|V(H_i)|=1$, then the unique vertex $v$ in $H_i$ is of course a $k$-king of $H_i$. Hence, in the following we assume that $|V(H_i)|\geq 2$.

If $H_i$ has no $k$-king, then since $v$ is a $k$-king of $Q$, there must exist a vertex $x\in V(H_i)\setminus \{v\}$ such that there is a $v-x$ path, denoted by $P_{vx}$, of length at most $k$ which uses at least one vertex from $V(Q)\setminus V(H_i)$. We claim that $u_i$ belongs to a cycle of length at most $k$ in $T$. To see this, we replace the vertices in $P_{vx}$ (in order) with their corresponding vertices in $T$. For example, we replace $v$ with $u_i$. Then we could get a closed walk of length at most $k$ which contains $u_i$ in $T$, this means that $u_i$ indeed belongs to a cycle of length at most $k$ in $T$. Hence, either $H_{i}$ has a $k$-king, or, $|V(H_{i})|\geq 2$ and $u_{i}$ belongs to a cycle of $T$ with length at most $k$.


\noindent{\bf Part (b)} The proof of {\em sufficiency} is not hard and we just prove the {\em necessity}. By Observation \ref{obs1}, all vertices of $T$ are $k$-kings of $T$. For any $i\in [t]$, if $|V(H_i)|=1$, then of course the unique vertex is a $k$-king of $H_i$. In the following we assume that $|V(H_i)|\geq 2$. If one vertex, say $u$, of $H_i$ is not $k$-king of $H_i$, then (since $u$ is a $k$-king of $Q$) there must exist a vertex $x\in V(H_i)\setminus \{u\}$ such that there is a $u-x$ path, denoted by $P_{ux}$, of length at most $k$ which uses at least one vertex from $V(Q)\setminus V(H_i)$. Then with a similar argument to that of \noindent{\bf Part (a)}, we have that $u_i$ belongs to a cycle of length at most $k$ in $T$.
\end{pf}


\subsection{$T$ is semicomplete}


The following result can be found in literature, see e.g. Theorem~2.2.9 of \cite{Bang-Jensen-Havet}.


\begin{thm}\label{thm02}
Every strong semicomplete digraph is vertex-pancyclic.
\end{thm}

\begin{thm}\label{thm01}\cite{Landau}
Every tournament has a king. More precisely, every vertex with maximum out-degree is a king.
\end{thm}

Bang-Jensen and Huang obtained the following result on the existence of 3-kings in a quasi-transitive digraph.

\begin{thm}\label{Bang-Huangking}\cite{Bang-Jensen-Huang2}
Let $D$ be a quasi-transitive digraph with a 3-king. If $D$ has no source, then $D$ has at least two 3-kings.
\end{thm}


For semicomplete compositions, we give a similar result to Theorem~\ref{Bang-Huangking}.

\begin{thm}\label{thmd}
Let $Q=T[H_1, \dots, H_t]$ be a semicomplete composition with a 3-king. If $T$ has no source, then $Q$ has at least two 3-kings. In particular, every strong semicomplete composition $Q=T[H_1, \dots, H_t]$ has at least two 3-kings, and a vertex $v\in H_i$ is a 3-king of $Q$ if and only if $u_i$ is a 3-king of $T$. 

\end{thm}
\begin{pf} We first prove the following claim.

{\bf Claim.} Every strong semicomplete composition $Q=T[H_1, \dots, H_t]$ has a 3-king. Furthermore, a vertex $v\in H_i$ is a 3-king of $Q$ if and only if $u_i$ is a 3-king of $T$.

{\bf Proof of the claim.} By Theorem~\ref{thm01}, there is a 3-king, say $u_i$, in $T$. If $|V(H_i)|=1$, then the unique vertex in $V(H_i)$ is also a 3-king of $Q$. Hence, we assume $|V(H_i)|\geq 2$. By Theorem~\ref{thm02}, $u_i$ belongs to a 3-cycle in $T$, so $Q$ has a 3-king by Theorem \ref{thma}.

If $u_i$ is a 3-king of $T$, then $v\in H_i$ can reach every vertex outside $H_i$ by a path of length at most three. By Theorem~\ref{thm02}, $u_i$ belongs to a 3-cycle in $T$, so $v$ can reach the remaining vertices of $H_i$ by a path of length at most three. The other direction is from Observation \ref{obs1}. \qed

\noindent{\bf Case 1:} $Q$ is strong.

By Observation~\ref{obs3}, $T$ is strong and of course has no source. By the above claim, $T$ has a 3-king and furthermore has at least two 3-kings by Theorem~\ref{Bang-Huangking}, therefore, $Q$ has at least two 3-kings.

\noindent{\bf Case 2:} $Q$ is not strong.

Since $Q$ has a 3-king, the initial strong component, say $Q'$, must be unique and contain all 3-kings. 
Furthermore, $Q'$ must contain vertices from at least two $H_i$. Otherwise, assume that $V(Q')\subseteq V(H_1)$, then since $Q'$ is the unique initial strong component, each vertex of $V(H_1)$ dominates every vertex of $V(Q)\setminus V(H_1)$ and there is no arc from $V(Q)\setminus V(H_1)$ to $V(H_1)$, this means that $u_1$ is a source in $T$, a contradiction. Observe that if $V(H_i)\cap V(Q')\neq\emptyset$, then $V(H_i)\subseteq V(Q')$, that is, $Q'$ is a union of some $H_i$. Without loss of generality, assume that $Q'$ contains vertices from $H_i~(1\leq i\leq t')$. Then $Q'$ is a strong semicomplete composition with the form $Q'=T'[H_1, \dots, H_{t'}]$ where $V(T')=\{u_i\mid 1\leq i\leq t'\}$ and $|T'|=t'\geq 2$. By the above claim, $Q'$ contains two 3-kings, say $x$ and $y$. Since $Q'$ is the unique initial strong component of $Q$, by the definition of semicomplete compositions, we have $x\rightarrow z$ and $y\rightarrow z$ for each $z\in V(Q)\setminus V(Q')$. Hence, both $x$ and $y$ are 3-kings of $Q$.


\end{pf}

\noindent{\bf Remark A}: According to the second paragraph of the proof for the claim in Theorem~\ref{thmd}, we actually can divide $H_1, \dots, H_t$ in a strong semicomplete composition $Q=T[H_1, \dots, H_t]$ into those for which all $v\in H_i$ are 3-kings (in this case, $u_i$ is a 3-king of $T$) and those for which no $v\in H_i$ is a 3-king.

Recall that in Theorem~\ref{Bang-Huangking}, Bang-Jensen and Huang \cite{Bang-Jensen-Huang2} proved that a quasi-transitive digraph $D$ with a 3-king contains at least two 3-kings provided that it has no source.
It is worth noting that we cannot replace the condition ``If $T$ has no source'' in Theorem~\ref{thmd} by ``If $Q$ has no source'', according to the following example: let $Q=T[H_1, \dots, H_t]$ be a semicomplete composition such that $u_i$ dominates (but is not dominated by) $u_j$ for any $1\leq i< j\leq t$, and $H_1$ is a digraph with vertex set $\{u_{1, j}\mid 1\leq j\leq 6\}$ and arc set $\{u_{1,j}u_{1,j+1}, u_{1,j+1}u_{1,j} \mid 1\leq j\leq 6\}$. It can be checked that $Q$ has no source but $u_{1,4}$ is the unique 3-king of $Q$.

The following result concerns the adjacency between 3-kings and non-kings.

\begin{thm}\label{thmc}
Let $Q=T[H_1, \dots, H_t]$ be a strong semicomplete composition. Then there is an arc between every 3-king and every non-king; moreover, for every non-king $u$, there exists a 3-king $v$ such that $d_Q(u, v)>3$ and $v$ dominates $u$.
\end{thm}
\begin{pf} By Theorem~\ref{thmd} and Remark A, there is an arc between every 3-king and every non-king.

Let $u$ be any non-king of $Q$. Let $H$ be the subdigraph of $Q$ induced by the set of all vertices $v$ such that $d_Q(u, v)>3$. Then we have $|V(H)|>0$, and furthermore, $V(H)=\bigcup_{j=1}^{t'}{V(H_{i_j})}~(1\leq t'\leq t)$ and each vertex of $V(H)$ dominates $u$ by the definition of a semicomplete composition (By Theorem~\ref{thm02}, $T$ is vertex-pancyclic and so $u$ can reach every other vertex of the same $H_i$ by a path of length at most three. Hence, $u\not\in V(H)$ and vertices in the same $H_i$ with $u$ do not belong to $V(H)$). Without loss of generality, we assume that $V(H)=\bigcup_{j=1}^{t'}{V(H_j)}~(1\leq t'\leq t)$. Let $T'$ be the subdigraph of $T$ induced by $\{u_i\mid 1\leq i\leq t'\}$. Clearly, $T'$ is semicomplete and hence has a king, say $u_1$, by Theorem~\ref{thm01}. Next we will prove the following claim.

{\bf Claim.} Every vertex of $H_1$ is a 3-king of $Q$.

{\bf Proof of the claim.} Let $x$ be a vertex of $H_1$. Suppose that $d_Q(x, y)>3$ for some vertex $y\in V(Q)$. For any vertex $x'\in V(H_1)\setminus \{x\}$, since $u_1$ belongs to a 3-cycle by Theorem~\ref{thm02}, we have $d_Q(x, x')\leq 3$. For any remaining vertex $y'$ of $H$, we clearly have $d_Q(x, y')\leq 2$ since $u_1$ is a king of $T'$. Hence, we must have $y\not\in H$, and so $d_Q(u, y)\leq 3$. Without loss of generality, assume that $d_Q(u, y)= 3$ and let $u, a, b, y$ be a $u-y$ path of length 3. There is an arc between $x$ and $a$ by the definition of a semicomplete composition. If $xa\in A(Q)$, then $x, a, b, y$ is an $x-y$ path of length 3, a contradiction. Otherwise, $ax\in A(Q)$, then $u, a, x$ is a $u-x$ path of length 2, which also produces a contradiction. Hence, the claim holds.\qed

Now we have that every vertex $x$ of $H_1$ is a 3-king of $Q$ such that $xu\in A(Q)$ and $d_Q(u,x)>3$, so the conclusion holds.
\end{pf}

Bang-Jensen and Huang \cite{Bang-Jensen-Huang2} proved that Theorem~\ref{thmc} holds for a quasi-transitive digraph with a 3-king, which means that the result may hold even for a non-strong quasi-transitive digraph (as long as it has a 3-king). However, for a non-strong semicomplete composition $Q$ with a 3-king, Theorem~\ref{thmc} may not hold. We just use the example after Theorem~\ref{thmd}: $Q$ is not strong and $u_{1,4}$ is the unique 3-king of $Q$; furthermore, there is no arc between $u_{1,4}$ and $u_{1,1}$, and of course $u_{1,4}$ does not dominate $u_{1,1}$.


We now give a sufficient condition under which a semicomplete composition can be established. It suffices to study strong semicomplete compositions with a non-king as the case that all vertices are 3-kings is trivial.

\begin{thm}\label{thme}
Let $Q=T[H_1, \dots, H_t]$ be a strong semicomplete composition. If $T$ has a strict 3-king and every 2-king of $T$ is dominated by some strict 3-king of $T$, then $Q$ can be established.
\end{thm}
\begin{pf}
Without loss of generality, let $\{u_i\mid 1\leq i\leq p\}$ and $\{u_i\mid p+1\leq i\leq q\}$ be the sets of strict 3-kings and 2-kings of $T$, respectively. We now construct a new digraph $Q'$ from $Q$ as follows:
let $V(Q')=V(Q)\cup \{v_i\mid 1\leq i\leq p\}$; for each $1\leq i\leq p$, $v_i$ completely dominates all vertices in $H_i$ and is completely dominated by all other vertices of $Q$; for any $1\leq i, j\leq p$, $v_i$ dominates $v_j$ if and only if $u_i$ dominates $u_j$.

Observe that $Q'=T'[H_1, \dots, H_t, v_1, \dots, v_p]$ is a strong semicomplete composition containing $Q$ as an induced subdigraph, where $V(T')=\{u_i\mid 1\leq i\leq t+p\}$ such that $u_{t+j}$ is corresponding to $v_j$ for $1\leq j\leq p$. Furthermore, our result holds by the following two claims.

{\bf Claim 1.} For each $1\leq i\leq p$, $v_i$ is a non-king of $Q'$.

{\bf Proof of Claim 1.} For each $1\leq i\leq p$, observe that all vertices in $H_i$ are strict 3-kings of $Q$, so they can not reach the vertices of some $H_j$ in two steps, hence $v_i$ can not reach the vertices of $H_j$ in three steps in $Q'$ and so $v_i$ is a non-king of $Q'$.\qed

{\bf Claim 2.} All vertices of $Q$ are 3-kings of $Q'$.

{\bf Proof of Claim 2.} Let $u\in V(Q)$. We need to consider the following three cases:

The first case is that $u\in \bigcup_{i=1}^p{V(H_i)}$. Without loss of generality, assume that $u\in V(H_1)$. Since $u_{1}$ is a 3-king of $T$, $u$ can reach any vertex $v\in V(Q)\setminus V(H_1)$ in at most three steps. For any vertex $v\in V(H_1)\setminus \{u\}$, the path $u, w, v_1, v$ is a $u-v$ path of length three, where $w\in V(Q)$ (note that the vertex $w$ exists since $u_{1}$ is a 3-king of $T$ and so has an out-neighbour in $V(T)\setminus \{u_1\}$). Moreover, $u$ dominates all new vertices.

The second case is that $u\in \bigcup_{i=p+1}^q{V(H_i)}$. Without loss of generality, assume that $u\in V(H_{p+1})$. Since $u_{p+1}$ is a 2-king of $T$, $u$ can reach any vertex $v\in V(Q)\setminus V(H_{p+1})$ in at most two steps. 
By the assumption, $u_{p+1}$ is dominated by some vertex, say $u_1$, from $\{u_i\mid 1\leq i\leq p\}$. Hence, for any vertex $v\in V(H_{p+1})\setminus \{u\}$, the path $u, v_1, w, v$ is a $u-v$ path of length three, where $w\in V(H_1)$. Moreover, $u$ dominates all new vertices.

Finally we consider the case that $u$ is a non-king of $Q$. If $v\in \bigcup_{i=1}^p{V(H_i)}$, then $u$ can reach $v$ by a path of the form $u, v_j, v$ if $v\in V(H_j)$. If $v\in \bigcup_{i=p+1}^q{V(H_i)}$,
then $u$ can reach $v$ by a path of the form $u, v_j, w, v$, where $w\in V(H_j)$ for some $1\leq j\leq p$. Such a path exists because every 2-king of $T$ is dominated by some strict 3-king of $T$ by the assumption. If $v$ is also a non-king, then $u$ can reach $v$ by a path of the form $u, v_j, w, v$, where $w\in V(H_j)$ for some $1\leq j\leq p$. To see why such a path exists, it suffices to notice that every non-king of $T$ must be dominated by some strict 3-king of $T$: Otherwise, there must exist some non-king $x$ which dominates all strict 3-kings of $T$. Consider any vertex $y\in V(T)$, if $y$ is a 2-king of $T$, then there exists an $x-y$ path in $T$: $x, x', y$, where $x'$ is a strict 3-king and its existence is from the assumption; if $y$ is also a non-king, then there exists an $x-y$ path in $T$: $x, x', y$, where $x'$ is a 3-king and its existence is from Theorem~\ref{thmc} as $T$ itself is a strong semicomplete composition, or an $x-y$ path in $T$: $x, x', x'', y$, where $x'$ is a strict 3-king and $x''$ is a 2-king. Hence, $x$ is a 3-king of $T$, which produces a contradiction.

Thus, in each case $u$ is a 3-king of $Q'$, so the claim holds.
\end{pf}

In the end of this section, we will discuss 4-kings in semicomplete compositions. In our argument, some results on semicomplete multipartite digraphs will be needed. 

For strong semicomplete multipartite digraphs, Gutin and Yeo obtained the following result on the number of 4-kings.

\begin{thm}\label{thm03}\cite{Gutin-Yeo}
Every strong semicomplete multipartite digraph with at least six vertices has at least five 4-kings.

\end{thm}

The following result on semicomplete bipartite digraphs was first stated by Wang and Zhang \cite{Wang-Zhang}, and was mentioned in \cite{GSHC} (see Theorem~8.6.7).

\begin{thm}\label{thm06}
Let $D$ be a semicomplete bipartite digraph with a unique initial strong component. If there is no 3-king in $D$, then there are at least eight 4-kings in $D$.
\end{thm}

For multipartite tournaments, Koh and Tan got the following result.

\begin{thm}\label{thm04}\cite{Koh-Tan, Koh-Tan2}
If a multipartite tournament has a unique initial strong component and no 3-king, then it has at least eight 4-kings.
\end{thm}

It is worth noting that many results for strong multipartite tournaments also hold for strong semicomplete multipartite digraphs, due to the following result by Volkmann.

\begin{thm}\label{thm05}\cite{Volkmann}
Every strong semicomplete $c$-partite digraph with $c \geq 3$ contains a spanning strong oriented subdigraph.
\end{thm}

Now we are ready to deal with the minimum number of $4$-kings in a strong semicomplete composition.

\begin{thm}\label{thmf}
Every strong semicomplete composition $Q=T[H_1, \dots, H_t]$ with at least six vertices has at least five 4-kings. Furthermore, if $Q$ has no 3-king, then it has at least eight 4-kings.

\end{thm}
\begin{pf} Let $Q'$ be the spanning subdigraph of $Q$ by deleting all arcs inside each $H_i$. 
By Observation~\ref{obs3}, $Q'$ is a strong semicomplete $t$-partite digraph with at least six vertices and so has at least five 4-kings by Theorem~\ref{thm03}. Hence, $Q$ has at least five 4-kings.

If $Q$ has no 3-king, then $Q'$ also has no 3-king by Observation~\ref{obs2}. In the following we will show that $Q'$ has at least eight 4-kings and then the result directly holds.
The case that $t=2$ holds by Theorem~\ref{thm06}. For the case that $t\geq 3$, by Theorem~\ref{thm05}, $Q'$ has a strong spanning subdigraph $Q''$ which is a strong $t$-partite tournament (without a 3-king), then $Q''$ (and so $Q'$) has at least eight 4-kings by Theorem~\ref{thm04}.
\end{pf}


\section{Kernels}\label{sec:Kernel}


Chv\'{a}tal and Lov\'{a}sz obtained the following result on a quasi-kernel in digraphs.

\begin{thm}\label{thm012}\cite{Chvatal-Lovasz}
Every digraph contains a quasi-kernel.
\end{thm}

Heard and Huang \cite{Heard-Huang} gave sufficient conditions to guarantee the existence of a pair of disjoint quasi-kernels in several classes of digraphs, including semicomplete digraphs.

\begin{thm}\cite{Heard-Huang}\label{thm011}
Every semicomplete digraph $D$ with no sink contains two vertices
$x, y$ such that $\{x\}$ and $\{y\}$ are both quasi-kernels of $D$.
\end{thm}

We now get a sufficient condition to guarantee the existence of a pair of disjoint quasi-kernels in semicomplete compositions.

\begin{thm}\label{thmb1}
Let $Q=T[H_1, \dots, H_t]$ be a semicomplete composition such that $T$ has no sink. Then $Q$ contains a pair of disjoint quasi-kernels. In particular, every strong semicomplete composition contains a pair of disjoint quasi-kernels.
\end{thm}
\begin{pf}
Let $Q=T[H_1, \dots, H_t]$ be a semicomplete composition such that $T$ has no sink. 
By Theorem~\ref{thm011}, there are two vertices, say $u_i, u_j$, such that $\{u_i\}$ and $\{u_j\}$ are both quasi-kernels of $T$. Let $Q_i$ and $Q_j$ be the quasi-kernels of $H_i$ and $H_j$, respectively (note that this can be guaranteed by Theorem~\ref{thm012}). It can be checked that $Q_i$ and $Q_j$ are disjoint quasi-kernels of $Q$.

If $Q=T[H_1, \dots, H_t]$ is a strong semicomplete composition, then $T$ is strong by Observation~\ref{obs3} and hence contains no sink. By the above argument, $Q$  contains a pair of disjoint quasi-kernels.
\end{pf}

In the remaining of this section, we turn our attention to the complexity of the problem {\sc $k$-Kernel} restricted to strong semicomplete compositions. We start with the following result which provides a necessary condition for a semicomplete composition to have a $k$-kernel when $k\geq 3$.

\begin{lem}\label{thmc1}
Let $Q=T[H_1, \dots, H_t]$ be a semicomplete composition. If $Q$ contains a $k$-kernel with $k\geq 3$, then there is a vertex $v\in V(Q)$ such that $\{v\}$ is a $(k-1)$-absorbent set of $Q-(V(H_i)\setminus \{v\})$ where $v\in V(H_i)$ for some $i\in [t]$.
\end{lem}
\begin{pf}
Let $K$ be a $k$-kernel of $Q$ and $v$ be any vertex of $K$. Observe that $K\subseteq V(H_i)$ for some $i\in [t]$. Let $Q'=Q-(V(H_i)\setminus \{v\})$ and $u$ be any vertex of $V(Q)\setminus V(H_i)$. Since $Q$ is a semicomplete composition, $u$ and $v$ must be adjacent in $Q$. If $u \rightarrow v$, then we are done. In the following we assume that $v \rightarrow u$.
By the definition of $K$, there must be $x\in K$ such that $d_Q(u, x)\leq k-1$. If $v=x$, then we are done. In the following we assume that $v\neq x$.

We claim that $d_Q(u, x)=k-1$. Indeed, if there is a $u-x$ path, denoted by $P_{u, x}$, in $Q$ of length at most $k-2$, then the path $vP_{u, x}$ is a $v-x$ path in $Q$ of length at most $k-1$, which contradicts with the $k$-independence of $K$. Hence, there is a $u-x$ path of length $k-1$ in $Q$: $u_0=u, u_1, \dots, u_{k-1}=x$. If there exists some $j\in [k-2]$ such that $u_j\in V(H_i)$ (we further assume $j$ is the smallest integer satisfying this property), then the path $u_0, u_1, \dots, u_{j-1}, v$ is a $u-v$ path in $Q'$ of length at most $k-1$. Otherwise, the path $u_0, u_1, \dots, u_{k-2}, v$ is a $u-v$ path in $Q'$ of length $k-1$. Therefore, in both cases, we conclude that $d_{Q'}(u, v)\leq k-1$. This completes the proof.
\end{pf}

Furthermore, when $k\geq 4$, we can characterize strong semicomplete compositions with a $k$-kernel.

\begin{lem}\label{thmd1}
Let $Q=T[H_1, \dots, H_t]$ be a strong semicomplete composition and $k\geq 4$. Then $Q$ has a $k$-kernel if and only if there is a vertex $v\in V(Q)$ such that $\{v\}$ is a $(k-1)$-absorbent set of $Q-(V(H_i)\setminus \{v\})$ where $v\in V(H_i)$ for some $i\in [t]$. 
\end{lem}
\begin{pf}
Suppose there is a vertex $v\in V(Q)$ such that $\{v\}$ is a $(k-1)$-absorbent set of $Q-(V(H_i)\setminus \{v\})$ where $i\in V(H_i)$ for some $i\in [t]$. Then $dist_Q(u, v)\leq dist_{Q'}(u, v)\leq k-1$ for any vertex $u\in V(Q)\setminus V(H_i)$, where $Q'=Q-(V(H_i)\setminus \{v\})$. By Theorem~\ref{thm02}, $u_i$ belongs to a cycle of length three in $T$, which means that $dist_Q(u, v)\leq 3$ for each $u\in V(H_i)\setminus \{v\}$. Hence, $\{v\}$ is a $(k-1)$-absorbent set of $Q$ and therefore is a $k$-kernel of $Q$. The other direction is clear by Lemma~\ref{thmc1}.
\end{pf}

The following result holds by the fact that $dist_T(u_j, u_i)=dist_{Q'}(u, v)$ for any $u\in V(H_j)$ where $j\in [t] \setminus \{i\}$, where $Q'=Q-(V(H_i)\setminus \{v\})$.

\begin{lem}\label{lem11}
Let $Q=T[H_1, \dots, H_t]$ be a digraph composition and $v\in V(H_i)$ for some $i\in [t]$. Then $\{v\}$ is a $k$-absorbent set of $Q-(V(H_i)\setminus \{v\})$ if and only if $\{u_i\}$ is a $k$-absorbent set of $T$.
\end{lem}


We can now prove the complexity of {\sc $k$-Kernel} restricted to strong semicomplete compositions for any integer $k\geq 2$.

\begin{thm}\label{thmd2}
Let $k\geq 2$ be an integer. The problem {\sc $k$-Kernel} restricted to strong semicomplete compositions is NP-complete when $k\in \{2,3\}$, and is polynomial time solvable when $k\geq 4$.
\end{thm}
\begin{pf}
Recall that the problem {\sc $k$-Kernel} for digraphs is NP-complete when $k\in \{2,3\}$ (see e.g. \cite{Chvatal, Hell-Hernandez-Cruz}). We first consider the case that $k=3$. Let $D$ be a digraph and construct a new digraph $Q$ by taking three copies of $D$, say $D_1, D_2, D_3$, and add all arcs from $D_1$ to $D_2$, all arcs from $D_2$ to $D_3$ and all arcs from $D_3$ to $D_1$, that is, $Q=\overrightarrow{C}_3[D, D, D]$. If $D$ contains a 3-kernel, say $K$, then $Q$ contains a 3-kernel as we just take the set $K$ in $D_1$. If $Q$ contains a 3-kernel, say $K'$, then we note that $K'\subseteq V(D_i)$ for some $i \in \{1,2,3\}$. This implies that $K'$ is also a 3-kernel of $D$. Therefore, $D$ contains a 3-kernel if and only if $Q$ does. As $Q$ is a strong semicomplete composition, this implies that the result holds for the case $k=3$. The case that $k=2$ can be proved similarly and we omit the details.

Now we consider the case that $k\geq 4$. Let $Q=T[H_1, \dots, H_t]$ be a strong semicomplete composition.
By Lemmas~\ref{thmd1} and~\ref{lem11}, $Q$ has a $k$-kernel if and only if $\{u_i\}$ is a $(k-1)$-absorbent set of $T$ for some $i\in [t]$. It suffices to show that we can decide in polynomial time if $\{u_i\}$ is a $(k-1)$-absorbent set of $T$, where $i\in [t]$. Indeed, we just compute the values of $dist_{T}(x, u_i)$ for all $x\in V(T)\setminus \{u_i\}$ and check if they are at most $k-1$. This can be done in polynomial time by the Dijkstra's algorithm (see e.g. Theorem~3.3.7 of \cite{Bang-Jensen-Gutin}).
\end{pf}


\vskip 1cm

\noindent {\bf Acknowledgement.} We are thankful to Professor Anders Yeo for
discussions on the complexity of {\sc 3-Kernel} problem restricted to strong semicomplete compositions. Yuefang Sun was supported by Zhejiang Provincial Natural Science Foundation (No. LY20A010013).
{"config": "arxiv", "file": "2006.05607.tex"}