On $m$-step competition graphs of bipartite tournaments

TL;DR

This paper characterizes the structure of m-step competition graphs of bipartite tournaments for all m ≥ 2, and calculates their competition index and period, advancing understanding of their properties.

Contribution

It provides a complete characterization of m-step competition graphs of bipartite tournaments and computes their competition index and period, which were previously unknown.

Findings

Complete characterization of m-step competition graphs for bipartite tournaments

Calculation of the competition index for bipartite tournaments

Determination of the competition period for bipartite tournaments

Abstract

In this paper, we completely characterize the $m$-step competition graph of a bipartite tournament for any integer $m \ge 2$. In addition, we compute the competition index and the competition period of a bipartite tournament.

Equations

$$(W_0,W_1,\dots ,W_{\zeta (D)})\text{mbox}and(D_0,D_1,\dots D_{\zeta (D)})$$

$$\bigcup _{0\le i\le (\zeta (D)-1)/2}{W}_{2i+1}\subset V_1\text{mbox}and\bigcup _{0\le i\le \zeta (D)/2}{W}_{2i}\subset V_2.$$

$$\mathrm{cindex}(D)=\{\begin{array}{ll}\zeta (D)& \text{mbox}if\zeta (D)=1\text{mbox}or|W_{\zeta (D)-1}|\ge 2;\\ \zeta (D)-1& \text{mbox}otherwise.\end{array}$$

$$K_{|V_1\setminus {\mathcal{W}}_m|}\cup K_{|V_2\setminus {\mathcal{W}}_m|}\cup I_{|{\mathcal{W}}_m|}.$$

$$N_D^{+}(x)\subset N_D^{-}(y)\text{and}{N}_D^{+}(y)\subset N_D^{-}(x).$$

$$X=\{v\in V_1\mid N_D^{+}(v)\subset N_D^{+}(x)\}\text{and}Y=\{v\in V_1\mid N_D^{+}(v)\subset N_D^{+}(y)\}.$$

$$X\cap Y=\{v\in V_1\mid N_D^{+}(v)\subset N_D^{+}(x)\cap N_D^{+}(y)\}.$$

$$X=N_D^{+}(x)\text{mbox}andY=N_D^{+}(y).$$

$$N_D^{+}(x)\subset N_D^{-}(y)\text{and}{N}_D^{+}(y)\subset N_D^{-}(x).$$

$$N_D^{-}(x)\cap N_D^{-}(y);N_D^{-}(x)\cap N_D^{+}(y)=Y;N_D^{+}(x)\cap N_D^{-}(y)=X.$$

$$Z:=\{z\in V_1\mid N_D^{+}(z)\ne X,N_D^{+}(z)\ne Y,\text{ and }{N}_D^{+}(z)\not\supset X\cup Y\}\ne \varnothing$$

$$\{\begin{array}{rlrlrlr}\mathrm{cindex}(D)\le 4& & \text{mbox}and& & \mathrm{cperiod}(D)=1& & \text{mbox}if\zeta (D)=0;\\ \mathrm{cindex}(D)\le 4& & \text{mbox}and& & \mathrm{cperiod}(D)\le 2& & \text{mbox}if\zeta (D)=1;\\ \mathrm{cindex}(D)=\zeta (D)& & \text{mbox}and& & \mathrm{cperiod}(D)=1& & \text{mbox}if\zeta (D)\ge 2.\end{array}$$

$$K_{|V(D_l)\cap V_i|}\cup I_{|V_i\setminus V(D_l)|}$$

$$K_{|V_i\setminus {\mathcal{W}}_m|}\cup I_{|V_i\cap {\mathcal{W}}_m|}$$

Full text

Keywords. $m$-step competition graph, bipartite tournament, competition index, competition period, sink elimination index, sink sequence

2010 Mathematics Subject Classification. 05C20, 05C75

1 Introduction

Given a digraph $D$, the competition graph $C(D)$ of $D$ has the same vertex set as $D$ and has an edge between vertices $u$ and $v$ if and only if there exists a common prey of $u$ and $v$ in $D$. If $(u,v)$ is an arc of a digraph $D$, then we call $v$ a prey of $u$ (in $D$) and call $u$ a predator of $v$ (in $D$). The notion of competition graph is due to Cohen [5] and has arisen from ecology. Competition graphs also have applications in coding, radio transmission, and modeling of complex economic systems. (See [13] and [14] for a summary of these applications.) Various variants of notion of competition graphs have been introduced and studied (see the survey articles by Kim [10] and Lundgren [11] for the variations which have been defined and studied by many authors since Cohen introduced the notion of competition graph).

The notion of $m$-step competition graph is one of the important variants and is defined as follows. Given a digraph $D$ and a positive integer $m$, a vertex $y$ is an $m$-step prey of a vertex $x$ if and only if there exists a directed walk from $x$ to $y$ of length $m$. Given a digraph $D$ and a positive integer $m$, the digraph $D^{m}$ has the vertex set same as $D$ and has an arc $(u,v)$ if and only if $v$ is an $m$-step prey of $u$. Given a positive integer $m$, the $m$-step competition graph of a digraph $D$, denoted by $C^{m}(D)$, has the same vertex set as $D$ and has an edge between vertices $u$ and $v$ if and only if there exists an $m$-step common prey of $u$ and $v$. The notion of $m$-step competition graph is introduced by Cho et al. [3] as a generalization of competition graph. By definition, it is obvious that $C^{1}(D)$ for a digraph $D$ is the competition graph $C(D)$. Since its introduction, it has been extensively studied (see for example [1, 2, 7, 8, 9, 12, 15]). Cho et al. [3] showed that for any digraph $D$ and a positive integer $m$, $C^{m}(D)=C(D^{m})$.

For the two-element Boolean algebra $\mathcal{B}=\{0,1\}$, $\mathcal{B}_{n}$ denotes the set of all $n\times n$ (Boolean) matrices over $\mathcal{B}$. Under the Boolean operations, we can define matrix addition and multiplication in $\mathcal{B}_{n}$. A graph $G$ is called the row graph of a matrix $A\in\mathcal{B}_{n}$ and denoted by $\mathcal{R}(A)$ if the rows of $A$ are the vertices of $G$, and two vertices are adjacent in $G$ if and only if their corresponding rows have a nonzero entry in the same column of $A$. This notion was studied by Greenberg et al. [6]. As noted in [6], the competition graph of a digraph $D$ is the row graph of its adjacency matrix.

Cho and Kim [4] introduced the notions of competition index and competition period of $D$ for a strongly connected digraph $D$, and Kim [9] extended these notions to a general digraph $D$. Consider the competition graph sequence $C^{1}(D)$, $C^{2}(D)$, $C^{3}(D)$, $\ldots$, $C^{m}(D)$, $\ldots$ for a digraph $D$. (Note that for a digraph $D$ and its adjacency matrix $A$, the graph sequence $C^{1}(D)$, $C^{2}(D)$, $\ldots$, $C^{m}(D)$, $\ldots$ is equivalent to the row graph sequence $\mathcal{R}(A)$, $\mathcal{R}(A^{2})$, $\ldots$, $\mathcal{R}(A^{m})$, $\ldots$.) Since the cardinality of the Boolean matrix set $\mathcal{B}_{n}$ is equal to a finite number $2^{n^{2}}$, there is a smallest positive integer $q$ such that $C^{q+i}(D)=C^{q+r+i}(D)$ (equivalently $\mathcal{R}(A^{q+i})=\mathcal{R}(A^{q+r+i})$) for some positive integer $r$ and all nonnegative integer $i$. Such an integer $q$ is called the competition index of $D$ and is denoted by cindex$(D)$. For $q=$cindex$(D)$, there is also a smallest positive integer $p$ such that $C^{q}(D)=C^{q+p}(D)$ (equivalently $\mathcal{R}(A^{q})=\mathcal{R}(A^{q+p})$). Such an integer $p$ is called the competition period of $D$ and is denoted by cperiod$(D)$.

Given a graph $G$, let $S\subset V(G)$ be any nonempty subset of vertices of $G$. The subgraph of $G$ induced by $S$, denoted by $G[S]$, is the graph whose vertex set is $S$ and whose edge set consists of all of the edges in $E(G)$ that have both endpoints in $S$. The same definition works for directed graphs.

In Section 2, we introduce notions of sink elimination index and sink sequence of a digraph and present some useful properties of bipartite tournaments related to $m$-step competition graphs in terms of them. In Section 3, we completely characterize the $m$-step competition graph of a bipartite tournament for any integer $m\geq 2$ and compute the competition index and the competition period of a bipartite tournament.

2 The sink elimination index and the sink sequence of a digraph

Given a digraph $D$, we call a vertex of outdegree zero a sink in $D$.

We define a nonnegative integer $\zeta(D)$ and sequences

[TABLE]

of subsets of $V(D)$ and subdigraphs of $D$, respectively, as follows. Let $D_{0}=D$ and $W_{0}$ be the set of sinks in $D$. If $W_{0}=V(D)$ or $W_{0}=\emptyset$, then let $\zeta(D)=0$. Otherwise, let $D_{1}=D_{0}-W_{0}$ and let $W_{1}$ be the set of sinks in $D_{1}$. If $W_{1}=V(D_{1})$ or $W_{1}=\emptyset$, then let $\zeta(D)=1$. Otherwise, let $D_{2}=D_{1}-W_{1}$ and let $W_{2}$ be the set of sinks in $D_{2}$. If $W_{2}=V(D_{2})$ or $W_{2}=\emptyset$, then let $\zeta(D)=2$. We continue in this way until we obtain $W_{k}=V(D_{k})$ or $W_{k}=\emptyset$ for some nonnegative integer $k$. Then we let $\zeta(D)=k$. We call $\zeta(D)$ the sink elimination index of $D$ and the sequence $(W_{0},W_{1},\ldots,W_{\zeta(D)})$ the sink sequence of $D$ (see Figure 1 for illustration) and the sequence $(D_{0},D_{1},\ldots,D_{\zeta(D)})$ the digraph sequence associated with the sink sequence.

By definition, it is easy to see that $V(D_{\zeta(D)})\neq\emptyset$ and $\bigcup_{i=0}^{\zeta(D)-1}{W_{i}}\cup V(D_{\zeta(D)})=V(D)$ for a digraph $D$. Therefore we have the following proposition.

Proposition 2.1**.**

For a digraph $D$, $W_{\zeta(D)}=V(D_{\zeta(D)})$ if and only if $\bigcup_{i=0}^{\zeta(D)}{W_{i}}=V(D)$.

Proposition 2.2**.**

A digraph $D$ is acyclic if and only if $W_{\zeta(D)}\neq\emptyset$.

Proof.

We note that $W_{\zeta(D)}=\emptyset$ if and only if $D_{\zeta(D)}$ has no sinks if and only if $D_{\zeta(D)}$ has a directed cycle. Therefore, if $W_{\zeta(D)}=\emptyset$, then $D_{\zeta(D)}$ has a directed cycle and so $D$ has a directed cycle. To show the converse, suppose that $D$ has a directed cycle $C$. Then any vertex on $C$ cannot belong to $W_{i}$ for any $i=0,\ldots,\zeta(D)$, so the vertices of $C$ belong to $V(D_{\zeta(D)})$. Thus $W_{\zeta(D)}\neq V(D_{\zeta(D)})$ and hence $W_{\zeta(D)}=\emptyset$ by definition. ∎

Let $D$ be a bipartite tournament with bipartition $(V_{1},V_{2})$ and $\zeta(D)\geq 1$, and let $(W_{0},\ldots,W_{\zeta(D)})$ and $(D_{0},D_{1},\ldots,D_{\zeta(D)})$ be the sink sequence and the digraph sequence associated with the sink sequence, respectively, of $D$. Fix $j\in\{0,\ldots,\zeta(D)\}$. By definition, $D_{j}$ is a bipartite tournament and $W_{j}$ is the set of sinks in $D_{j}$. Therefore $W_{j}$ is included in exactly one of partite sets of $D_{j}$. Since the partite sets of $D_{j}$ are included in $V_{1}$ and $V_{2}$, respectively, $W_{j}$ is included in exactly one of $V_{1}$ and $V_{2}$.

Now we state the following proposition.

Proposition 2.3**.**

Let $D$ be a bipartite tournament with bipartition $(V_{1},V_{2})$. In addition, let $(W_{0},\ldots,W_{\zeta(D)})$ be the sink sequence of $D$ with $\zeta(D)\geq 1$. Then, $\bigcup_{0\leq i\leq{\zeta(D)}/2}{W_{2i}}$ is included in one of the bipartite sets while $\bigcup_{0\leq i\leq({\zeta(D)}-1)/2}{W_{2i+1}}$ is included in the other partite set. Furthermore, $D$ is acyclic if and only if $\bigcup_{0\leq i\leq{\zeta(D)}/2}{W_{2i}}$ and $\bigcup_{0\leq i\leq({\zeta(D)}-1)/2}{W_{2i+1}}$ themselves are the bipartite sets.

Proof.

Let $(D_{0},D_{1},\ldots,D_{\zeta(D)})$ be the digraph sequence associated with $(W_{0},\ldots,W_{\zeta(D)})$. Since $\zeta(D)\geq 1$, $\emptyset\subsetneq W_{0}\subsetneq V(D)$. Suppose that $W_{j}\subset V_{1}$ for some $j\in\{0,\ldots,\zeta(D)-1\}$. Since $D$ is a bipartite tournament, $D_{j}$ is a bipartite tournament and so $W_{j+1}\subset V_{2}$. Similarly, if $W_{j}\subset V_{2}$ for some $j\in\{0,\ldots,\zeta(D)-1\}$, then $W_{j+1}\subset V_{1}$. Thus we have shown that $\bigcup_{0\leq i\leq{\zeta(D)}/2}{W_{2i}}$ is included in one of the bipartite sets while $\bigcup_{0\leq i\leq({\zeta(D)}-1)/2}{W_{2i+1}}$ is included in the other partite set.

The “furthermore” part may be justified as follows. Without loss of generality, we may assume that

[TABLE]

Now,

[TABLE]

∎

Proposition 2.4**.**

Let $D$ be a bipartite tournament with $\zeta(D)\geq 1$ and $(W_{0},\ldots,W_{\zeta(D)})$ be the sink sequence of $D$. Then any directed walk with an initial vertex in $W_{i}$ has length at most $i$ in $D$ for $i=0,\ldots,\zeta(D)-1$. Furthermore, if $D$ is acyclic, then even a directed walk with an initial vertex in $W_{\zeta(D)}$ has length at most $\zeta(D)$ in $D$.

Proof.

Fix $i\in\{0,\ldots,\zeta(D)-1\}$ and take a directed walk $W$ in $D$ with an initial vertex $v_{i}$ in $W_{i}$. Let $\alpha$ be the length of $W$. Then there are $\alpha$ terms after $v_{i}$ in the sequence of vertices on $W$. Suppose to the contrary that there exists a vertex $w$ on $W$ that is distinct from $v_{i}$ and belongs to $V(D_{\zeta(D)})$. Without loss of generality, we may assume that $w$ is the first vertex on $W$ that belongs to $V(D_{\zeta(D)})$. Then the vertex right before $w$ on $W$ belongs to $W_{t}$ for some $t\in\{0,1,\ldots,\zeta(D)\}$, which contradicts the definition of sink sequence. Therefore each vertex on $W$ belongs to $W_{j}$ for some $j\in\{0,1,\ldots,\zeta(D)\}$. Now suppose that there exist two consecutive vertices $w_{1}$ and $w_{2}$ on $W$ such that $(w_{1},w_{2})$ is an arc on $W$ and $w_{1}\in W_{j}$ and $w_{2}\in W_{k}$ for some positive integers $j$ and $k$ satisfying $j\leq k$. Then $(w_{1},w_{2})$ belongs to $D_{j}$ by the definition of digraph sequence associated with $(W_{0},\ldots,W_{\zeta(D)})$. However, the vertices in $W_{j}$ have outdegree zero in $D_{j}$ and we reach a contradiction. Therefore, if $(x,y)$ is an arc on $W$ for some vertices $x$ and $y$ on $W$, then $x\in W_{j}$ and $y\in W_{k}$ for some positive integers $j$ and $k$ satisfying $j-k\geq 1$. Therefore, if the terminus belongs to $W_{l}$ for some nonnegative integer $l$, then $i-l\geq\alpha$. Since $l\geq 0$, we have $\alpha\leq i$ and so the first part of the lemma statement is valid.

If $D$ is acyclic, then, by Proposition 2.2, $W_{\zeta(D)}\neq\emptyset$. We may take a directed walk with an initial vertex in $W_{\zeta(D)}$. Then, by the above argument, the length of the walk is at most $\zeta(D)$. ∎

Proposition 2.5**.**

Let $D$ be a bipartite tournament with bipartition $(V_{1},V_{2})$. In addition, let $\zeta(D)\geq 1$ and $(W_{0},\ldots,W_{\zeta(D)})$ be the sink sequence of $D$, and $\mathcal{W}_{k}=\bigcup_{i=0}^{k-1}W_{i}$ for $1\leq k\leq\zeta(D)+1$. Then the following are true:

• (1)

A vertex in $W_{2k^{\prime}+1}$ is an out-neighbor of each vertex in $V_{2}\setminus\mathcal{W}_{2k^{\prime}+1}$ in $D$ for $k^{\prime}=0,\ldots,\left\lfloor{{\zeta(D)}/{2}}\right\rfloor-1$.

• (2)

A vertex in $W_{2k^{\prime}}$ is an out-neighbor of each vertex in $V_{1}\setminus\mathcal{W}_{2k^{\prime}}$ in $D$ for $k^{\prime}=1,\ldots,\left\lceil{{\zeta(D)}/{2}}\right\rceil-1$.

• (3)

If $v_{i}\in W_{i}$ for each $i=0,\ldots,l$ ($1\leq l<\zeta(D)$), then there exists a directed path $v_{l}\rightarrow v_{l-1}\rightarrow\cdots\rightarrow v_{1}\rightarrow v_{0}$. Furthermore, if $D$ is acyclic, then there is an arc from any vertex in $W_{\zeta(D)}$ to any vertex in $W_{\zeta(D)-1}$.

Proof.

Since $\zeta(D)\geq 1$, $W_{0}\neq\emptyset$. Since $D$ is a bipartite tournament, $W_{0}\subset V_{1}$ or $W_{0}\subset V_{2}$. Without loss of generality, we may assume $W_{0}\subset V_{2}$. Then, by Proposition 2.3, $W_{i}\subset V_{1}$ for each odd integer $0\leq i\leq\zeta(D)$ and $W_{i}\subset V_{2}$ for each even integer $0\leq i\leq\zeta(D)$. Take a vertex $v_{i}\in W_{i}$ for each $i=0,\ldots,\zeta(D)-1$. If $D$ is acyclic, then $W_{\zeta(D)}\neq\emptyset$ by Proposition 2.2 and we may take $v_{\zeta(D)}\in W_{\zeta(D)}$. Fix $k^{\prime}\in\left\{0,\ldots,\left\lfloor{{\zeta(D)}/{2}}\right\rfloor-1\right\}$. Then, since $D$ is a bipartite tournament, $v_{2k^{\prime}+1}$ is an out-neighbor of each vertex in $V_{2}\setminus\mathcal{W}_{2k^{\prime}+1}$ in $D$. Therefore there exists an arc from $v_{2k^{\prime}+2j+2}$ to $v_{2k^{\prime}+1}$ for each nonnegative integer $j$ with $2k^{\prime}+2j+2\leq\zeta(D)$. Similarly, for each $k^{\prime}=1,\ldots,\left\lceil{{\zeta(D)}/{2}}\right\rceil-1$, $v_{2k^{\prime}}$ is an out-neighbor of each vertex in $V_{1}\setminus\mathcal{W}_{2k^{\prime}}$ in $D$, so there exists an arc from $v_{2k^{\prime}+2j+1}$ to $v_{2k^{\prime}}$ for each nonnegative integer $j$ with $2k^{\prime}+2j+1\leq\zeta(D)$. Since $v_{i}$ was arbitrarily chosen in $W_{i}$, we have shown (1) and (2). We also have shown the following:

• •

there exists an arc from each vertex in $W_{t}$ to each vertex in $W_{s}$ for positive integers $0\leq s<t<\zeta(D)$ whenever $t-s$ is an odd integer;

• •

if $D$ is acyclic, there exists an arc from each vertex in $W_{\zeta(D)}$ to each vertex in $W_{s}$ for positive integer $0\leq s<\zeta(D)$ whenever $\zeta(D)-s$ is an odd integer.

The statement (3) immediately follows from these two facts. ∎

3 A characterization of the $m$-step competition graph of a bipartite tournament

In this section, we completely characterize the $m$-step competition graph of a bipartite tournament for any integer $m\geq 2$. In addition, we compute the competition index and the competition period of a bipartite tournament.

We first present fundamental properties of $m$-step competition graphs of bipartite tournaments.

Proposition 3.1**.**

For a bipartite tournament $D$ with bipartition $(V_{1},V_{2})$, there is no edge joining a vertex in $V_{1}$ and a vertex in $V_{2}$ in $C^{m}(D)$ for any positive integer $m$.

Proof.

For a vertex in $V_{1}$, a vertex in $V_{1}$ can be only $2k$-step prey for a positive integer $k$ and a vertex in $V_{2}$ can be only $(2k^{\prime}-1)$-step prey for a positive integer $k^{\prime}$ while, for a vertex in $V_{2}$, a vertex in $V_{1}$ can be only $(2l-1)$-step prey for a positive integer $l$ and a vertex in $V_{2}$ can be only $2l^{\prime}$-step prey for a positive integer $l^{\prime}$. Therefore a vertex in $V_{1}$ and a vertex in $V_{2}$ cannot have an $m$-step common prey for any positive integer $m$. ∎

Proposition 3.2**.**

Let $D$ be a bipartite tournament with bipartition $(V_{1},V_{2})$. If $C^{M}(D)$ has an edge for a positive integer $M$, then so does $C^{m}(D)$ for any positive integer $m\leq M$.

Proof.

If $M=1$, then $m=1$ and the statement is trivially true. Suppose $M\geq 2$. Let $xy$ be an edge in $C^{M}(D)$. Then $x$ and $y$ belong to the same part by Proposition 3.1. Without loss of generality, we may assume that $x$ and $y$ belong to $V_{1}$. In addition, $x$ and $y$ have an $M$-step common prey $z$ in $D$. Then there exist a directed $(x,z)$-walk $P$ and a directed $(y,z)$-walk $Q$ of length $M$ in $D$. Let $x_{1}$ and $y_{1}$ be the vertices in $D$ such that $(x,x_{1})$ and $(y,y_{1})$ are the arcs on $P$ and $Q$, respectively. If $x_{1}$ and $y_{1}$ are distinct, then $z$ is an $(M-1)$-step common prey of $x_{1}$ and $y_{1}$ and so $x_{1}$ and $y_{1}$ are adjacent in $C^{(M-1)}(D)$. If $x_{1}$ and $y_{1}$ are the same, then the vertex immediately following $z$ on $P$ is an $(M-1)$-step common prey of $x$ and $y$ and so $x$ and $y$ are adjacent in $C^{(M-1)}(D)$. Therefore there is an edge in $C^{(M-1)}(D)$. If $M\geq 3$, then we may repeat this argument to show that there is an edge in $C^{(M-2)}(D)$. In this way, we may show that there is an edge in $C^{m}(D)$ for any positive integer $m\leq M$. ∎

The following corollary is the contrapositive of Proposition 3.2.

Corollary 3.3**.**

Let $D$ be a bipartite tournament with bipartition $(V_{1},V_{2})$. If $C^{M}(D)$ is an edgeless graph for a positive integer $M$, then so does $C^{m}(D)$ for any positive integer $m\geq M$.

Proposition 3.4**.**

Let $D$ be a bipartite tournament with no sinks. If two vertices are adjacent in $C^{M}(D)$ for a positive integer $M$, then they are also adjacent in $C^{m}(D)$ for any positive integer $m\geq M$.

Proof.

Let $x$ and $y$ are adjacent in $C^{M}(D)$. Then $x$ and $y$ have an $M$-step common prey $z$ in $D$. By the hypothesis, $z$ has an out-neighbor $w$ in $D$. Then $w$ is an $(M+1)$-step common prey of $x$ and $y$. Hence $x$ and $y$ are adjacent in $C^{(M+1)}(D)$. We may repeat this argument to show that $x$ and $y$ are adjacent in $C^{(M+2)}(D)$. In this way, we may show that $x$ and $y$ are adjacent in $C^{m}(D)$ for any positive integer $m\geq M$. ∎

Proposition 3.5**.**

Let $D$ be a bipartite tournament with bipartition $(V_{1},V_{2})$. In addition, let $\zeta(D)\geq 1$ and $(W_{0},\ldots,W_{\zeta(D)})$ be the sink sequence of $D$, and $\mathcal{W}_{k}=\bigcup_{i=0}^{k-1}W_{i}$ for $1\leq k\leq\zeta(D)+1$. Then each of $V_{1}\setminus\mathcal{W}_{m}$ and $V_{2}\setminus\mathcal{W}_{m}$ forms a clique in $C^{m}(D)$ for a positive integer $m<\zeta(D)$.

Proof.

Since $1\leq m<\zeta(D)$, $V_{1}\setminus\mathcal{W}_{m}\neq\emptyset$ and $V_{2}\setminus\mathcal{W}_{m}\neq\emptyset$. Take a vertex $v$ in $V_{1}\setminus\mathcal{W}_{m}$. Then, since $\mathcal{W}_{m-1}\subset\mathcal{W}_{m}$, $v\in V_{1}\setminus\mathcal{W}_{m-1}$. If $m$ is even, there exists an arc from $v$ to any vertex in $W_{m}$ by Proposition 2.5(2) and so any vertex in $W_{1}$ is an $m$-step prey of $v$ by (3) of the same proposition. If $m$ is odd, there exists an arc from $v$ to any vertex in $W_{m-1}$ by Proposition 2.5(2) since $v\in V_{1}\setminus\mathcal{W}_{m-1}$, and so any vertex in $W_{0}$ is an $m$-step prey of $v$. Since $v$ is arbitrarily chosen, a vertex in $W_{0}$ or a vertex in $W_{1}$ is an $m$-step prey of every vertex in $V_{1}\setminus\mathcal{W}_{m}$ depending upon the parity of $m$. Thus $V_{1}\setminus\mathcal{W}_{m}$ forms a clique in $C^{m}(D)$. By applying a similar argument, we may show that $V_{2}\setminus\mathcal{W}_{m}$ forms a clique in $C^{m}(D)$. ∎

Next, we characterize the $m$-step competition graph of an acyclic bipartite tournament and compute the competition index and the competition period of an acyclic bipartite tournament.

For given two graphs $G_{1}$ and $G_{2}$, we call the graph having the vertex set $V(G_{1})\cup V(G_{2})$ and the edge set $E(G_{1})\cup E(G_{2})$ the union of $G_{1}$ and $G_{2}$ and denote it by $G_{1}\cup G_{2}$. Unless otherwise mentioned, $G_{1}\cup G_{2}$ stands for the union of vertex-disjoint graphs $G_{1}$ and $G_{2}$.

Theorem 3.6**.**

Let $D$ be an acyclic bipartite tournament having bipartition $(V_{1},V_{2})$, $(W_{0},\ldots,W_{\zeta(D)})$ be the sink sequence of $D$, and $\mathcal{W}_{k}=\bigcup_{i=0}^{k-1}W_{i}$ for $1\leq k\leq\zeta(D)+1$. Then, for a positive integer $m$, the following are true:

• (1)

$C^{m}(D)$* is an empty graph if $m>\zeta(D)$;*

• (2)

$C^{m}(D)$* is isomorphic to $K_{|W_{\zeta(D)}|}\cup I_{|\mathcal{W}_{m}|}$ if $m=\zeta(D)$;*

• (3)

$C^{m}(D)$* is isomorphic to $K_{|V_{1}\setminus\mathcal{W}_{m}|}\cup K_{|V_{2}\setminus\mathcal{W}_{m}|}\cup I_{|\mathcal{W}_{m}|}$ if $m<\zeta(D)$;*

• (4)

$\mathrm{cperiod}(D)=1$;

• (5)

$\mathrm{cindex}(D)=\zeta(D)+1$* if $|W_{\zeta(D)}|\geq 2$; otherwise,*

[TABLE]

Proof.

Suppose $\zeta(D)=0$. Then, by definition, $W_{0}=V(D)$ or $W_{0}=\emptyset$. Since $D$ is a bipartite tournament, $W_{0}\neq V(D)$. Yet, since $D$ is acyclic, $W_{0}\neq\emptyset$ by Proposition 2.2. Therefore we have reached a contradiction. Therefore $\zeta(D)\geq 1$. Suppose $m>\zeta(D)$. Since $D$ is acyclic, $\bigcup_{i=0}^{\zeta(D)}W_{i}=V(D)$ by the furthermore part of Proposition 2.3 and so no vertex in $D$ has an $m$-step prey in $D$ by Proposition 2.4. Therefore $C^{m}(D)$ is an empty graph and so the statement (1) is true. Then, by the definition of competition period, the statement (4) is immediately true. Now suppose that $m\leq\zeta(D)$. By Proposition 2.4, no vertex in $\mathcal{W}_{m}$ has an $m$-step prey in $D$ and so every vertex in $\mathcal{W}_{m}$ is an isolated vertex in $C^{m}(D)$.

Suppose $m=\zeta(D)$. Since $D$ is acyclic, $W_{\zeta(D)}\neq\emptyset$ by Proposition 2.2. By the furthermore part of Proposition 2.3, $V(D)=W_{\zeta(D)}\cup\mathcal{W}_{m}$. Then, since $\zeta(D)\geq 1$, $W_{\zeta(D)}$ forms a clique in $C^{m}(D)$ by Proposition 2.5. Since we have shown that every vertex in $\mathcal{W}_{m}$ is an isolated vertex in $C^{m}(D)$, $C^{m}(D)$ is isomorphic to $K_{|W_{\zeta(D)}|}\cup I_{|\mathcal{W}_{m}|}$.

Now suppose $m<\zeta(D)$. Then, by Proposition 3.5, each of $V_{1}\setminus\mathcal{W}_{m}$ and $V_{2}\setminus\mathcal{W}_{m}$ forms a clique in $C^{m}(D)$. Moreover, by Proposition 3.1, there is no edge between $K_{|V_{1}\setminus\mathcal{W}_{m}|}$ and $K_{|V_{2}\setminus\mathcal{W}_{m}|}$. Since every vertex in $\mathcal{W}_{m}$ is an isolated vertex in $C^{m}(D)$, $C^{m}(D)$ is isomorphic to

[TABLE]

Thus the statement (3) is true.

By (1), $\mathrm{cindex}(D)\leq\zeta(D)+1$. In addition, by (1), it is sufficient to show that $C^{i}(D)$ is not an empty graph in order to prove that $\mathrm{cindex}(D)\geq i+1$ for a positive integer $i$. If $|W_{\zeta(D)}|\geq 2$, then $C^{\zeta(D)}(D)$ has edges joining each pair of vertices in $W_{\zeta(D)}$ by Proposition 2.5 and so $\mathrm{cindex}(D)=\zeta(D)+1$. Now suppose $|W_{\zeta(D)}|\leq 1$. Since $D$ is acyclic, $W_{\zeta(D)}\neq\emptyset$ by Proposition 2.2. Therefore $|W_{\zeta(D)}|=1$. If $\zeta(D)=1$, then $C(D)$ is empty and so $\mathrm{cindex}(D)=1=\zeta(D)$. Consider the case $\zeta(D)\geq 2$. If $|W_{\zeta(D)-1}|\geq 2$, then $C^{\zeta(D)-1}(D)$ has edges joining each pair of vertices in $W_{\zeta(D)-1}$ and so, by the supposition $|W_{\zeta(D)}|=1$ and Proposition 2.4, $\mathrm{cindex}(D)=\zeta(D)$. Suppose that $|W_{\zeta(D)-1}|=1$. Then $C^{\zeta(D)-2}(D)$ has at least one edge joining a vertex in $W_{\zeta(D)}$ and a vertex in $W_{\zeta(D)-2}$ by Proposition 3.5 and so $\mathrm{cindex}(D)\geq\zeta(D)-1$. By Proposition 2.4, the vertices in $\mathcal{W}_{\zeta(D)-1}$ are isolated in $C^{\zeta(D)-1}(D)$. By Proposition 3.1, any vertex in $W_{\zeta(D)}$ and any vertex in $W_{\zeta(D)-1}$ are not adjacent in $C^{\zeta(D)-1}(D)$. Thus, by the suppositions $|W_{\zeta(D)}|=|W_{\zeta(D)-1}|=1$, $C^{\zeta(D)-1}(D)$ is an empty graph. Then, by Corollary 3.3, $C^{\zeta(D)}(D)$ is an empty graph. Hence we may conclude that $\mathrm{cindex}(D)=\zeta(D)-1$. ∎

In the following, we shall characterize the $m$-step competition graph of a bipartite tournament having a directed cycle and compute the competition index and the competition period of a bipartite tournament having a directed cycle. To do so, we need the following lemmas.

Lemma 3.7**.**

Let $D$ be a bipartite tournament having bipartition $(V_{1},V_{2})$ without sinks and $G$ be the $m$-step competition graph of $D$ for an integer $m\geq 2$. Then, $G[V_{i}]$ is a complete graph or (not necessarily disjoint) union of two complete graphs for each $i=1,2$.

Proof.

By symmetry, it is sufficient to show that the statement is true for $G_{1}:=G[V_{1}]$. If $G_{1}$ is complete, then the statement is trivially true. Therefore we may assume that $G_{1}$ is not complete. Then there exist two nonadjacent vertices, say $x$ and $y$, in $G_{1}$. Since $D$ has no sink, $x$ and $y$ cannot have a common out-neighbor in $D$ by Proposition 3.4. Then, since $D$ is a bipartite tournament,

[TABLE]

Let $X$ and $Y$ be the sets defined by

[TABLE]

Since $x\in X$ and $y\in Y$, $X\neq\emptyset$ and $Y\neq\emptyset$. Then,

[TABLE]

Since $N^{+}_{D}(v)\neq\emptyset$ for any $v\in V_{1}$ and $N^{+}_{D}(x)\cap N^{+}_{D}(y)=\emptyset$, we have $X\cap Y=\emptyset$. By (2), every vertex in $X$ (resp. $Y$) has $y$ (resp. $x$) as a $2$-step prey, so each of $X$ and $Y$ forms a clique in $C^{m}(D)$ by Proposition 3.4. An $m$-step prey of a vertex in $X$ (resp. $Y$) is an $m$-step prey of $x$ (resp. $y$) by definition. Since $x$ and $y$ are nonadjacent in $G_{1}$, $x$ and $y$ do not have an $m$-step common prey in $D$. Therefore any vertex in $X$ and any vertex in $Y$ do not have an $m$-step common prey in $D$ and thus are not adjacent in $G_{1}$. Now, if $X\cup Y=V_{1}$, then the statement is immediately true.

Suppose $X\cup Y\neq V_{1}$. Then $Z:=V_{1}\setminus(X\cup Y)\neq\emptyset$. Take a vertex $z\in Z$. By definition, there exist $v$ and $w$ in $N^{+}_{D}(z)$ satisfying $v\notin N^{+}_{D}(x)$ and $w\notin N^{+}_{D}(y)$. Since $D$ is a bipartite tournament, $v\in N^{-}_{D}(x)$ and $w\in N^{-}_{D}(y)$. Thus $x$ and $y$ are $2$-step prey of $z$. Since $x$ (resp. $y$) is a $2$-step prey of every vertex in $Y$ (resp. $X$), every vertex in $X\cup Y$ is adjacent to $z$ in $G_{1}$ by Proposition 3.4. Since $z$ was arbitrarily chosen, every vertex in $Z$ is adjacent to every vertex in $X\cup Y$. Since every vertex in $Z$ has $x$ as a $2$-step prey, $Z$ forms a clique in $G_{1}$ by Proposition 3.4. Thus we may conclude that each of $Z\cup X$ and $Z\cup Y$ forms a clique in $G_{1}$. As we have shown that any vertex in $X$ and any vertex in $Y$ are not adjacent in $G_{1}$, $G_{1}$ is a union of two complete graphs. Hence we have shown that $G_{1}$ is a complete graph or a union of two complete graphs. ∎

Lemma 3.8**.**

Let $D$ be a bipartite tournament having bipartition $(V_{1},V_{2})$ without sinks. Then $\mathrm{cperiod}(D)=1$ and $\mathrm{cindex}(D)\leq 4$.

Proof.

Cho and Kim [4] showed that a digraph without sinks has competition period $1$, so $\mathrm{cperiod}(D)=1$. In the following, we shall show $\mathrm{cindex}(D)\leq 4$. Suppose that there exist two vertices $x$ and $y$ such that $x$ and $y$ are adjacent in $C^{M}(D)$ for some positive integer $M$. Without loss of generality, we may assume that $x$ and $y$ belong to $V_{1}$ by Proposition 3.1. By Proposition 3.4, it is sufficient to show that $x$ and $y$ are adjacent in $C^{m}(D)$ for some positive integer $m\leq 4$. Let

[TABLE]

By the hypothesis, $X\neq\emptyset$ and $Y\neq\emptyset$. If $X\cap Y\neq\emptyset$, then $x$ and $y$ are adjacent in $C(D)$ and we are done. Consider the case $X\cap Y=\emptyset$. Then, since $D$ is a bipartite tournament,

[TABLE]

Moreover, since $X\cap Y=\emptyset$, $V_{2}$ is a disjoint union of the following sets:

[TABLE]

Suppose to the contrary that, for each $z\in V_{1}$, $N_{D}^{+}(z)=X$ or $N_{D}^{+}(z)=Y$ or $N_{D}^{+}(z)\supset X\cup Y$. Then, $V_{1}$ is a disjoint union of $Z_{1}:=\{z\in V_{1}\mid N_{D}^{+}(z)=X\}$, $Z_{2}:=\{z\in V_{1}\mid N_{D}^{+}(z)=Y\}$, and $V_{1}\setminus(Z_{1}\cup Z_{2})$. Then the only possible $m$-step prey of $x$ or $y$ are vertices in $Y$, vertices in $X$, vertices in $Z_{1}$, or vertices in $Z_{2}$. To see why, suppose there exists a $(u,v)$-directed walk $W$ for some $u\in V$ and $v\in V_{1}\setminus(Z_{1}\cup Z_{2})$ (resp. $v\in V_{2}\setminus(X\cup Y)$). By definition, the vertex $v_{1}$ right before $v$ on $W$ belongs to $V_{2}\setminus(X\cup Y)$ (resp. $V_{1}\setminus(Z_{1}\cup Z_{2})$). Then, the vertex right before $v_{1}$ on $W$ belongs to $V_{1}\setminus(Z_{1}\cup Z_{2})$ (resp. $V_{2}\setminus(X\cup Y)$). Therefore any vertex in $V_{1}\setminus(Z_{1}\cup Z_{2})$ or $V_{2}\setminus(X\cup Y)$ is reachable only from a vertex in $V_{1}\setminus(Z_{1}\cup Z_{2})$ or $V_{2}\setminus(X\cup Y)$. Thus we have shown that the only possible $m$-step prey of $x$ or $y$ are vertices in $Y$, vertices in $X$, vertices in $Z_{1}$, or vertices in $Z_{2}$. Yet, a vertex in $Y$ (resp. $X$) can be only $(4k_{1}+3)$-step (resp. $(4k_{1}+1)$-step) prey of $x$ while it can be only $(4k_{2}+1)$-step (resp. $(4k_{2}+3)$-step) prey of $y$, and a vertex in $Z_{1}$ (resp. a vertex in $Z_{2}$) can be only $4k_{3}$-step (resp. $(4k_{3}+2)$-step) prey of $x$ while it can be only $(4k_{4}+2)$-step (resp. $4k_{4}$-step) prey of $y$ for nonnegative integers $k_{1}$, $k_{2}$, $k_{3}$, and $k_{4}$. Therefore there are no $m$-step common prey of $x$ and $y$ for any integer $m\geq 1$, and we reach a contradiction. Thus there exists $z\in V_{1}$ such that $N_{D}^{+}(z)\neq X$, $N_{D}^{+}(z)\neq Y$, and $N_{D}^{+}(z)\not\supset X\cup Y$. Hence

[TABLE]

Suppose that there is $z\in Z$ such that $N_{D}^{+}(z)\not\supset X$ and $N_{D}^{+}(z)\not\supset Y$. Then we may take two vertices $w_{1}\in X\setminus N_{D}^{+}(z)$ and $w_{2}\in Y\setminus N_{D}^{+}(z)$. Since $D$ is a bipartite tournament, $(w_{1},z)\in A(D)$ and $(w_{2},z)\in A(D)$. Thus $z$ is a $2$-step common prey of $x$ and $y$ in $D$ and we are done. Now suppose that $N_{D}^{+}(z)\supset X$ or $N_{D}^{+}(z)\supset Y$ for any $z\in Z$ and fix $z\in Z$. Without loss of generality, we may assume $N_{D}^{+}(z)\supset X$. Then, since $N_{D}^{+}(z)\neq X$ and $N_{D}^{+}(z)\not\supset X\cup Y$ by the definition of $Z$, $N_{D}^{+}(z)\setminus X\neq\emptyset$ and $N_{D}^{+}(z)\not\supset Y$. Now we may take $w_{4}\in N_{D}^{+}(z)\setminus X$ and $w_{5}\in Y\setminus N_{D}^{+}(z)$. Since $D$ is a bipartite tournament, $(w_{4},x)\in A(D)$ and $(w_{5},z)\in A(D)$. Since $X\neq\emptyset$, we may take $w_{3}\in X$. By the way, since $X\cap Y=\emptyset$, $(w_{3},y)\in A(D)$ and $(w_{5},x)\in A(D)$. Thus $x\rightarrow w_{3}\rightarrow y\rightarrow w_{5}\rightarrow x$ and $y\rightarrow w_{5}\rightarrow z\rightarrow w_{4}\rightarrow x$ are directed walks in $D$. Hence $x$ and $y$ have $x$ as a $4$-step common prey in $D$ and we are done. ∎

The upper bound given in Lemma 3.8 is sharp as seen by the digraph $D$ given in Figure 2. The digraph $D$ has no sinks, $C^{3}(D)\neq C^{4}(D)$, $C^{m}(D)=C^{4}(D)$ for any integer $m\geq 4$. Therefore the competition index of $D$ is $4$.

Theorem 3.9**.**

Let $D$ be a bipartite tournament with bipartition $(V_{1},V_{2})$ which has a directed cycle and $G_{m}$ be the $m$-step competition graph of $D$ for an integer $m\geq 2$. Then, $G_{m}[V_{i}]$ is a disjoint union of complete graphs among which at most two are nontrivial, or, by deleting the isolated vertices from $G_{m}[V_{i}]$ if any, we obtain a non-disjoint union of two complete graphs for each $i=1,2$. Moreover,

[TABLE]

Proof.

Let $(W_{0},W_{1},\ldots,W_{\zeta(D)})$ be the sink sequence of $D$ and $G_{m,i}=G_{m}[V_{i}]$ for $i=1,2$. Suppose $\zeta(D)=0$. Then, since $D$ has a directed cycle, $W_{0}=\emptyset$ by Proposition 2.2. Therefore the theorem statement is true by Lemmas 3.7 and 3.8.

Now suppose $\zeta(D)\geq 1$. By Proposition 2.3, $\bigcup_{0\leq i\leq{\zeta(D)}/2}{W_{2i}}$ is included in one of the bipartite sets while $\bigcup_{0\leq i\leq({\zeta(D)}-1)/2}{W_{2i+1}}$ is included in the other partite set. By symmetry, we may assume that $\bigcup_{0\leq i\leq{\zeta(D)}/2}{W_{2i}}\subset V_{2}$ and $\bigcup_{0\leq i\leq({\zeta(D)}-1)/2}{W_{2i+1}}\subset V_{1}$. Let $(D_{0},D_{1},\ldots,D_{\zeta(D)})$ be the digraph sequence associated with $(W_{0},\ldots,W_{\zeta(D)})$.

Case 1. $\zeta(D)=1$. Then $(W_{0},W_{1})$ is the sink sequence of $D$ with $W_{0}\subset V_{2}$ and $W_{1}\subset V_{1}$. Since $\zeta(D)=1$, $W_{1}=V(D_{1})$ or $W_{1}=\emptyset$. By the hypothesis, $D$ has a directed cycle, so $W_{1}=\emptyset$ by Proposition 2.2. Thus every vertex in $D_{1}$ has outdegree at least one. Obviously, each vertex in $W_{0}$ is isolated in $G_{m}$. Since $W_{1}\neq V(D_{1})$, $W_{0}\subsetneq V_{2}$. Thus $D_{1}$ is a bipartite tournament with bipartition $(V_{1},V_{2}\setminus W_{0})$.

Consider the case where $m$ is odd. Then, since every vertex in $D_{1}$ has outdegree at least one, every vertex in $V_{1}$ has an $(m-1)$-step prey in $V_{1}$ in $D_{1}$ and thus in $D$. Therefore each vertex in $W_{0}$ is an $m$-step prey of each vertex in $V_{1}$ in $D$. Thus $G_{m,1}$ is complete. Furthermore, since every vertex in $D_{1}$ has outdegree at least one, the subgraph $H$ of $C^{m}(D_{1})$ induced by $V_{2}\setminus W_{0}$ is a complete graph or a union of two complete graphs by Lemma 3.7. Since the subgraph of $C^{m}(D)$ induced by $V_{2}\setminus W_{0}$ is $G_{m,2}-W_{0}$, $H$ is a subgraph of $G_{m,2}-W_{0}$. If an edge $e$ belongs to $G_{m,2}-W_{0}$, then the end vertices of $e$ have an $m$-step common prey in $D$ which belongs to $V_{1}$. Moreover, since each vertex in $W_{0}$ has outdegree zero, any directed walk of length $m$ does not contain a vertex in $W_{0}$ as an interior vertex. Therefore the adjacency of two vertices belonging to $G_{m,2}-W_{0}$ is inherited to $H$. Then, since $H$ is a spanning subgraph of $G_{m,2}-W_{0}$, $H=G_{m,2}-W_{0}$. Thus $G_{m,1}$ is complete, the vertices in $W_{0}$ are isolated in $G_{m}$, and $G_{m,2}-W_{0}$ is a complete graph or a union of two complete graphs. By applying a similar argument for the case in which $m$ is even, we may show that $G_{m,2}-W_{0}$ is complete, the vertices in $W_{0}$ are isolated in $G_{m}$, and $G_{m,1}$ is a complete graph or a union of two complete graphs. Hence the first part of the theorem is true.

Now we show $\mathrm{cperiod}(D)\leq 2$. Suppose that $x$ and $y$ are adjacent in $G_{M}$ for some $x,y\in V_{2}$ and some odd integer $M\geq 3$. Then $x$ and $y$ have an $M$-step common prey in $D$. Since $M$ is odd, the $M$-step common prey of $x$ and $y$ in $D$ are contained in $V_{1}$. Since every vertex of $D_{1}$ has outdegree at least one, $x$ and $y$ have an $(M+2)$-step common prey in $D$. Thus $x$ and $y$ are adjacent in $G_{M+2}$. By applying a similar argument, we may show that if $x$ and $y$ are adjacent in $G_{M}$ for $x,y\in V_{1}$ for some even integer $M\geq 2$, then $x$ and $y$ are adjacent in $G_{M+2}$. Thus we have shown that

• ($\star$)

if two vertices are adjacent in $G_{m}$ for $m\geq 2$, then they are adjacent in $G_{m+2}$.

Note that $G_{N,1}$ is complete and the vertices in $W_{0}$ are isolated in $G_{N}$ for any odd integer $N\geq 3$ and $G_{N^{\prime},2}-W_{0}$ is complete and the vertices in $W_{0}$ are isolated in $G_{N^{\prime}}$ for any even integer $N^{\prime}\geq 2$. Hence $\mathrm{cperiod}(D)\leq 2$.

In the following, we compute the competition index of $D$. In a previous argument, we have shown that for any odd integer $N\geq 3$, $G_{N,1}$ is complete and for any even integer $N^{\prime}$, $G_{N^{\prime},2}-W_{0}$ is complete. Now take two vertices $x$ and $y$ in $V_{1}$ which are adjacent in $G_{M}$ for some even integer $M\geq 4$. Then $x$ and $y$ have an $M$-step common prey $z$ in $D$. Since $M$ is even, $z$ belongs to $V_{1}$. By the definition of $W_{0}$, neither any $(x,z)$-directed walk nor any $(y,z)$-directed walk contains a vertex in $W_{0}$, so $z$ is an $M$-step common prey of $x$ and $y$ in $D_{1}$. Since $D_{1}$ has no sinks, $\mathrm{cperiod}(D_{1})=1$ and $\mathrm{cindex}(D_{1})\leq 4$ by Lemma 3.8. Thus $x$ and $y$ are adjacent in $C^{4}(D_{1})$. Since $C^{4}(D_{1})$ is a subgraph of $G_{4}$, $x$ and $y$ are adjacent in $G_{4}$. By ($\star$), $x$ and $y$ are adjacent in $G_{N^{\prime}}$ for any even integer $N^{\prime}\geq 4$. By applying a similar argument, we may show that if two vertices in $V_{2}$ are adjacent in $G_{M}$ for some odd integer $M\geq 5$, then they are adjacent in $G_{N}$ for any odd integer $N\geq 5$. Hence we have shown that $\mathrm{cindex}(D)\leq 4$.

Case 2. $\zeta(D)\geq 2$. Let $l=\zeta(D)$. Since $D$ has a directed cycle, $W_{l}=\emptyset$ by Proposition 2.2 and so each vertex in $D_{l}$ has outdegree at least one. Since $D$ contains a directed cycle and $D_{l}$ is a bipartite tournament, $|V(D_{l})\cap V_{1}|\geq 2$. Now we take two distinct vertices $u$ and $v$ in $V(D_{l})\cap V_{1}$. Since each vertex in $D_{l}$ has outdegree at least one, $u$ (resp. $v$) has an $(m-1)$-step prey $u^{\prime}$ (resp. $v^{\prime}$) in $V(D_{l})$. Obviously, $u^{\prime}$ and $v^{\prime}$ belong to $V_{1}$ if $m$ is odd; to $V_{2}$ if $m$ is even. If $u^{\prime}$ and $v^{\prime}$ belong to $V_{1}$ (resp. $V_{2}$), then they have a common prey $w$ in $W_{0}$ (resp. $W_{1}$) by the definition of sink sequence. Therefore $w$ is an $m$-step common prey of $u$ and $v$ in $D$ and so $V(D_{l})\cap V_{1}$ forms a clique in $G_{m}$. By a similar argument, it can be shown that $V(D_{l})\cap V_{2}$ forms a clique in $G_{m}$.

For notational convenience, let $\mathcal{W}_{k}=\bigcup_{i=0}^{k-1}W_{i}$ for $1\leq k\leq\zeta(D)+1$.

Subcase 1. $m\geq l$. Then, by Proposition 2.4, the vertices in $\mathcal{W}_{l}$ are isolated in $G_{m}$. Since we have shown that $V(D_{l})\cap V_{i}$ forms a clique in $G_{m}$, $G_{m,i}$ is isomorphic to

[TABLE]

for each $i=1,2$.

Subcase 2. $2\leq m\leq l-1$. Then each of $V_{1}\setminus\mathcal{W}_{m}$ and $V_{2}\setminus\mathcal{W}_{m}$ forms a clique in $G_{m}$ by Proposition 3.5. By Proposition 2.4, the vertices in $\mathcal{W}_{m}$ are isolated in $G_{m}$. Thus $G_{m,i}$ is isomorphic to

[TABLE]

for each $i=1,2$.

By the conclusion deduced in Subcase 1, $\mathrm{cperiod}(D)=1$ and $\mathrm{cindex}(D)\leq l$. Suppose that $l$ is even. Then $W_{l-1}\subset V_{1}$. By Proposition 2.4, the vertices in $W_{l-1}$ are isolated in $G_{l}$. However, by Proposition 3.5, $V_{1}\setminus\mathcal{W}_{l-1}$ forms a clique in $G_{l-1}$. Since $W_{l-1}\subset\left(V_{1}\setminus\mathcal{W}_{l-1}\right)$, $W_{l-1}$ forms a clique in $G_{l-1}$. Thus $G_{l}\neq G_{l-1}$ and so $\mathrm{cindex}(D)\geq l$. We may apply similar argument to show that $\mathrm{cindex}(D)\geq l$ for an odd $l$. Hence $\mathrm{cindex}(D)=l$. ∎

4 Acknowledgement

This research was supported by the National Research Foundation of Korea(NRF) funded by the Korea government(MEST) (NRF-2017R1E1A1A03070489) and by the Korea government(MSIP) (2016R1A5A1008055).