Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs

TL;DR

This paper investigates the existence of arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs, providing conditions for strong and semicomplete cases and polynomial-time decision algorithms.

Contribution

It generalizes previous results by characterizing when compositions of digraphs have good pairs at every vertex and offers polynomial-time algorithms for semicomplete compositions.

Findings

Every strong digraph composition with each $n_i \\ge 2$ has a good pair at every vertex.

Characterization of semicomplete compositions with a good pair, extending prior work.

Polynomial-time decision procedure for the existence of a good pair in semicomplete compositions.

Abstract

A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair of arc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ 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 $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ When $T$ is arbitrary, we obtain the following result: every strong digraph composition $Q$ in which $n_i\ge 2$ for every $1\leq i\leq t$, has a good pair at every vertex of $Q.$ The condition of $n_i\ge 2$ in this result cannot be relaxed. When $T$ is semicomplete, we characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

Equations

$$A(Q)={\cup }_{i=1}^{t}A(H_i)\cup \{u_{i{j}_i}{u}_{p{q}_p}\mid u_i{u}_p\in A(T),1\le j_i\le n_i,1\le q_p\le n_p\}.$$

Full text

[TABLE]

When $T$ is arbitrary, we obtain the following result: every strong digraph composition $Q$ in which $n_{i}\geq 2$ for every $1\leq i\leq t$, has a good pair at every vertex of $Q.$ The condition of $n_{i}\geq 2$ in this result cannot be relaxed. When $T$ is semicomplete, we characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (J. Graph Theory, 1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.

Keywords: branching; semicomplete digraph; digraph composition.

AMS subject classification (2010): 05C20, 05C70, 05C76, 05C85.

1 Introduction

We use a standard digraph terminology and notation as in [4, 5]. A digraph $D=(V,A)$ is strongly connected (or strong) if there exists a path from $x$ to $y$ and a path from $y$ to $x$ in $D$ for every pair of distinct vertices $x,y$ of $D$. An out-tree (in-tree, respectively) rooted at a vertex $r$ is an orientation of a tree such that the in-degree (out-degree, respectively) of every vertex but $r$ equals one. An out-branching ( in-branching, respectively) in a digraph $D$ is a spanning subgraph of $D$ which is out-tree (in-tree, respectively). It is well-known and easy to show [4, 5] that a digraph has an out-branching (in-branching, respectively) rooted at $r$ if and only if $D$ has a unique initial strong connectivity component (terminal strong connectivity component, respectively) and $r$ belongs to this component. Out-branchings and in-branchings when they exist can be found in linear-time using, say, depth-first search from the root.

Edmonds [11] characterized digraphs with $k$ arc-disjoint out-branchings rooted at a specified vertex $r.$ Furthermore, there exists a polynomial algorithm for finding $k$ arc-disjoint out-branchings with a given root $r$ if they exist [4]). However, it is NP-complete to decide whether a digraph $D$ has a pair of arc-disjoint out-branching and in-branching rooted at $r,$ which was proved by Thomassen, see [1]. Following [9] we will call such a pair a good pair rooted at $r$. Note that a good pair forms a strong spanning subgraph of $D$ and thus if $D$ has a good pair, then $D$ is strong. The problem of the existence of a good pair was studied for tournaments and their generalizations, and characterizations (with proofs implying polynomial-time algorithms for finding such a pair) were obtained in [1] for tournaments, [7] for quasi-transitive digraphs and [9] for locally semicomplete digraphs. Also, Bang-Jensen and Huang [7] showed that if $r$ is adjacent to every vertex of $D$ (apart from itself) then $D$ has a good pair rooted at $r$.

In this paper, we study the existence of good pairs for digraph compositions. Let $T$ be a digraph with $t$ vertices $u_{1},\dots,u_{t}$ and let $H_{1},\dots H_{t}$ be digraphs such that $H_{i}$ has vertices $u_{i,j_{i}},\ 1\leq j_{i}\leq n_{i}.$ Then the composition $Q=T[H_{1},\dots,H_{t}]$ is a digraph with vertex set $\{u_{i,j_{i}}\mid 1\leq i\leq t,1\leq j_{i}\leq n_{i}\}$ and arc set

[TABLE]

Digraph compositions generalize some families of digraphs. In particular, semicomplete compositions, which are compositions where $T$ is a semicomplete digraph, generalize strong quasi-transitive digraphs as every strong quasi-transitive digraph is a strong semicomplete composition in which $H_{i}$ is either a one-vertex digraph or a non-strong quasi-transitive digraph. To see that strong compositions form a significant generalization of strong quasi-transitive digraphs, observe that the Hamiltonian cycle problem is polynomial-time solvable for quasi-transitive digraphs [13], but NP-complete for strong compositions (see, e.g., [6]). 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., [15]. While digraph compositions has been used since 1990s to study quasi-transitive digraphs and their generalizations, see, e.g., [3, 4, 12], the study of digraph decompositions in their own right was initiated only recently in [16].

In the next section, we obtain the following somewhat surprising result: every strong digraph composition $Q$ in which $n_{i}\geq 2$ for every $i\in[t]$, has a good pair rooted at every vertex of $Q.$ The condition of $n_{i}\geq 2$ in this result cannot be relaxed. Indeed, the characterization of quasi-transitive digraphs with a good pair [7] provides an infinite family of strong quasi-transitive digraphs which have no good pair rooted at some vertices. In Section 3, we characterize semicomplete compositions with a good pair generalizing the corresponding result in [7]. This allows us to decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex. In Section 4, we discuss some open problems and a recent related result.

Let $p$ and $q$ be integers. Then $[p..q]:=\{p,p+1,\dots,q\}$ if $p\leq q$ and $\emptyset$, otherwise. In particular, if $p>0$, $[p]$ will be a shorthand for $[1..p].$

2 Compositions of digraphs: $T$ arbitrary

A digraph $D=(V,A)$ has a strong arc decomposition if $A$ has two disjoint sets $A_{1}$ and $A_{2}$ such that both $(V,A_{1})$ and $(V,A_{2})$ are strong. Sun et al. [16] obtained sufficient conditions for a digraph composition to have a strong arc decomposition. In particular, they proved the following:

Theorem 2.1

Let $T$ be a digraph with $t$ vertices ($t\geq 2$) and let $H_{1},\dots,H_{t}$ be digraphs. Then $Q=T[H_{1},\dots,H_{t}]$ has a strong arc decomposition if $T$ has a Hamiltonian cycle and one of the following conditions holds:

• •

$t$* is even and $n_{i}\geq 2$ for every $i=1,\dots,t;$*

• •

$t$* is odd, $n_{i}\geq 2$ for every $i=1,\dots,t$ and at least two distinct subgraphs $H_{i}$ have arcs;*

• •

$t$* is odd and $n_{i}\geq 3$ for every $i=1,\dots,t$ apart from one $i$ for which $n_{i}\geq 2$.*

Lemma 2.2

Let $Q=T[H_{1},\dots,H_{t}],$ where $t\geq 2.$ If $T$ has a Hamiltonian cycle and $H_{1},\dots,H_{t}$ are arbitrary digraphs, each with at least two vertices, then $Q$ has a good pair at any root $r$.

Proof: For the case that $t$ is even, by Theorem 2.1, $Q$ has has a pair of arc-disjoint strong spanning subgraphs $Q_{1}$ and $Q_{2}$. Observe that in $Q_{1}$ ($Q_{2}$, respectively), we can find an out-branching (in-branching, respectively) at $r$ (in polynomial time), as desired.

Now we assume that $t$ is odd. Without loss of generality, let $u_{1,1}$ be the root. Let $T^{\prime}_{1}$ be the path $u_{1,1}u_{2,1}\dots u_{t,1}u_{1,2}u_{2,2}\dots u_{t,2}$, and let $T^{\prime}_{2}$ be the in-tree rooted at $u_{1,1}$ with arc set $\{u_{i,2}u_{i+1,1}\mid 1\leq i\leq t-1\}\cup\{u_{i,1}u_{i+1,2}\mid 2\leq i\leq t-1\}\cup\{u_{t,1}u_{1,1},u_{t,2}u_{1,1}\}$. By definition, $V(T^{\prime}_{1})=V(T^{\prime}_{2})=\{u_{i,j}\mid 1\leq i\leq t,1\leq j\leq 2\}$. For any vertex $u_{i,j}$ with $1\leq i\leq t$ and $j\geq 3$, we add the arcs $u_{i-1,1}u_{i,j}$ and $u_{i,j}u_{i+1,1}$ to $T^{\prime}_{1}$ and $T^{\prime}_{2}$, respectively. Note that here $u_{0,1}=u_{t,1}$ and $u_{t+1,1}=u_{1,1}$. Observe that the resulting two subgraphs form a pair of out-branching and in-branching rooted at $u_{1,1}$, which are arc-disjoint. $\Box$

We will use the following decomposition of strong digraphs. An ear decomposition of a digraph $D$ is a sequence $\mathcal{P}=(P_{0},P_{1},P_{2},\cdots,P_{t})$, where $P_{0}$ is a cycle or a vertex and each $P_{i}$ is a path, or a cycle with the following properties:

$(a)$ $P_{i}$ and $P_{j}$ are arc-disjoint when $i\neq j$.

$(b)$ For each $i\in[0..t]$, let $D_{i}$ denote the digraph with vertices $\bigcup_{j=0}^{i}{V(P_{j})}$ and arcs $\bigcup_{j=0}^{i}{A(P_{j})}$. If $P_{i}$ is a cycle, then it has precisely one vertex in common with $V(D_{i-1})$. Otherwise the end vertices of $P_{i}$ are distinct vertices of $V(D_{i-1})$ and no other vertex of $P_{i}$ belongs to $V(D_{i-1})$.

$(c)$ $\bigcup_{j=0}^{t}{A(P_{j})}=A(D)$.

The following result is well-known, see, e.g., [4].

Theorem 2.3

Let $D$ be a digraph with at least two vertices. Then $D$ is strong if and only if it has an ear decomposition. Furthermore, if $D$ is strong, every cycle can be used as a starting cycle $P_{0}$ for an ear decomposition of $D$, and there is a linear-time algorithm to find such an ear decomposition.

Lemma 2.4

Let $Q=T[\overline{K_{2}},\dots,\overline{K_{2}}],$ where $|V(T)|=t\geq 2$ and $\overline{K_{2}}$ is the digraph with two vertices and no arcs. If $T$ is strong, then $Q$ has a good pair at any root $r$.

Proof: Without loss of generality, let $r=u_{1,1}$. Since $T$ is strong, $u_{1}$ belongs to some cycle $C$ in $T$. By Theorem 2.3, $T$ has an ear decomposition $\mathcal{P}=(P_{0},P_{1},P_{2},\cdots,P_{p})$, such that $P_{0}=C$ is the starting cycle. Let $G_{i}$ denote the subgraph of $G$ with vertices $\bigcup_{j=0}^{i}{V(P_{j})}$ and arcs $\bigcup_{j=0}^{i}{A(P_{j})}$.

We will prove the lemma by induction on $i\in\{0,1,\dots,p\}$. For the base step, by Lemma 2.2, the subgraph $P_{0}[\overline{K_{2}},\dots,\overline{K_{2}}]$ has a good pair rooted at $u_{1}$. For the inductive step, assume that $G_{i}[\overline{K_{2}},\dots,\overline{K_{2}}]$ has a pair of arc-disjoint out-branching $T_{1}^{\prime}$ and in-branching $T_{2}^{\prime}$ rooted at $r$. Without loss of generality, let $P_{i+1}=u_{s}u_{s+1}\dots u_{\ell}$. The following argument will be divided into two cases according to whether $P_{i+1}$ is a cycle.

Case 1: $P_{i+1}$ is a cycle.

In this case $u_{s}=u_{\ell}\in V(G_{i})$. By Lemma 2.2, in the subgraph $P_{i+1}[\overline{K_{2}},\dots,\overline{K_{2}}]$, there is a pair of arc-disjoint out-branching $T_{1}^{\prime\prime}$ and in-branching $T_{2}^{\prime\prime}$ rooted at $u_{s,1}$. Let $T_{1}=T_{1}^{\prime}\cup T_{1}^{\prime\prime}$ and $T_{2}=T_{2}^{\prime}\cup T_{2}^{\prime\prime}$. Observe that $T_{1}$ is an out-branching and $T_{2}$ is an in-branching rooted at $r$ in $G_{i+1}[\overline{K_{2}},\dots,\overline{K_{2}}].$ Since $P_{i+1}[\overline{K_{2}},\dots,\overline{K_{2}}]$ and $G_{i}[\overline{K_{2}},\dots,\overline{K_{2}}]$ are arc-disjoint, $T_{1}$ and $T_{2}$ are arc-disjoint.

Case 2: $P_{i+1}$ is a path.

In this case, $u_{s},u_{\ell}\in V(G_{i})$ and $s\neq\ell$. We just consider the case that $\ell-s\geq 2$ since the remaining case is trivial (no need to change the current pair of out- and in-branchings). Let $T_{1}$ be the union of $T_{1}^{\prime}$ and the two paths $u_{s,i}u_{s+1,i}\dots u_{\ell-1,i}$ where $1\leq i\leq 2$. Let $T_{2}$ be the union of $T_{2}^{\prime}$ and the two paths $u_{s,1}u_{s+1,2}u_{s+2,1}u_{s+3,2}\dots u_{\ell,i}$ and $u_{s,2}u_{s+1,1}u_{s+2,2}u_{s+3,1}\dots u_{\ell,j},$ where $\{i,j\}=\{1,2\}.$ Observe that $T_{1}$ is an out-branching and $T_{2}$ is an in-branching rooted at $r$ in $G_{i+1}[\overline{K_{2}},\dots,\overline{K_{2}}]$, moreover, they are arc-disjoint.

Thus, by induction, we are done. $\Box$

Theorem 2.5

Let $Q=T[H_{1},\dots,H_{t}],$ where $t\geq 2.$ If $T$ is strong and $H_{1},\dots,H_{t}$ are arbitrary digraphs, each with at least two vertices, then $Q$ has a good pair at any root $r$. Furthermore, this pair can be found in polynomial time.

Proof: Without loss of generality, let $r=u_{1,1}$. Let $Q^{\prime}$ be the subgraph of $Q$ induced by the vertex set $\{u_{i,j}\mid 1\leq i\leq t,1\leq j\leq 2\}$. In $Q^{\prime}$ delete arcs between vertices $u_{i,1}$ and $u_{i,2}$ for every $i\in[t].$ By Lemma 2.4, $Q^{\prime}$ contains a pair of arc-disjoint out-tree $T_{1}^{\prime}$ and in-tree $T_{2}^{\prime}$ rooted at $r$. By definition of out-tree, there is an arc $u_{p_{i},q_{i}}u_{i,2}$ in $T_{1}^{\prime}$ for every $i\in[t].$ For every $i\in[t]$ and $j\in[3..n_{i}],$ add $u_{p_{i},q_{i}}u_{i,j}$ to $T_{1}^{\prime}.$ This results in an out-branching $T_{1}.$ By definition of in-tree, there is an arc $u_{i,2}u_{a_{i},b_{i}}$ in $T_{2}^{\prime}$ for every $i\in[t].$ For every $i\in[t]$ and $j\in[3..n_{i}],$ add $u_{i,j}u_{a_{i},b_{i}}$ to $T_{2}^{\prime}.$ This results in an in-branching $T_{2}.$ Observe that $T_{1}$ and $T_{2}$ are arc-disjoint since $T^{\prime}_{1}$ and $T^{\prime}_{2}$ are arc-disjoint and the added arcs have heads and tails from $\{u_{i,j}\mid 1\leq i\leq t,3\leq j\leq n_{i}\},$ respectively, in the arcs added to $T^{\prime}_{1}$ and $T^{\prime}_{2},$ respectively. Note that the proofs of Theorem 2.1, Lemmas 2.2 and 2.4, and this theorem are constructive and can be converted into polynomial-time algorithms. This fact and the polynomial-time algorithm of Theorem 2.3 imply that $T_{1}$ and $T_{2}$ can be constructed in polynomial time. $\Box$

3 Compositions of digraphs: $T$ semicomplete

We use $N^{-}(v)$ ($N^{+}(v)$, respectively) to denote the set of all in-neighbours (out-neighbours, respectively) of a vertex $v$ in a digraph $D$.

The next result was obtained by Bang-Jensen and Huang [7].

Theorem 3.1

Let $D$ be a strong digraph and $r$ a vertex of $D$ such that $V(D)=\{r\}\cup N^{-}(r)\cup N^{+}(r)$. There is a polynomial-time algorithm to decide whether $D$ has a good pair at $r$.

For a path $P=x_{1}x_{2}\dots x_{p}$ and $1\leq i\leq j\leq p$, let $P[x_{i},x_{j}]:=x_{i}x_{i+1}\dots x_{j}.$ We now prove the following result on semicomplete compositions which generalizes a similar result for quasi-transitive digraphs by Bang-Jensen and Huang [7].

Theorem 3.2

A strong semicomplete composition $Q$ has a good pair rooted at $r$ if and only if $Q^{\prime}=Q[\{r\}\cup N^{-}(r)\cup N^{+}(r)]$ has a good pair rooted at $r$.

Proof: Let $Q=T[H_{1},\dots,H_{t}]$ and $A=V(Q)\setminus V(Q^{\prime})$. Without loss of generality, assume that $r\in V(H_{1})$. By definitions of a semicomplete composition and $Q^{\prime}$, we have $A=V(H_{1})\setminus\{r\}$.

Assume that $Q^{\prime}$ has a good pair rooted at $r$, an out-branching $B^{\prime+}_{r}$ and an in-branching $B^{\prime-}_{r}$. Starting with $B^{\prime+}_{r}$, we can construct an out-branching $B^{+}_{r}$ in $Q$ as follows. Let $v$ be a vertex such that $vr\in B^{\prime-}_{r}$. Then add the arc $vu$ to $B^{\prime+}_{r}$ for each $u\in A.$ Similarly, starting with $B^{\prime-}_{r}$, we could construct an in-branching $B^{-}_{r}$ in $Q$ as follows: for each $u\in A$, add the arc $uv^{\prime}$ to $B^{\prime-}_{r},$ where $rv^{\prime}\in B^{\prime+}_{r}$. Observe that $B^{+}_{r}$ and $B^{-}_{r}$ are arc-disjoint, as desired.

Now we prove the other direction. Assume that $Q$ has a good pair, an out-branching $B^{+}_{r}$ and an in-branching $B^{-}_{r}$, rooted at $r$. If $B^{+}_{r}[V(Q^{\prime})]$ and $B^{-}_{r}[V(Q^{\prime})]$ are branchings, then we are done. Otherwise, we will obtain an in-branching (out-branching, respectively) from $B^{-}_{r}$ ($B^{+}_{r},$ respectively) using the following procedure.

Choose a maximal path $P$ of $B^{-}_{r}$ to $r$, which contains a vertex $w\in A$, and assume that $w$ is furthest from $r$ among vertices in $A\cap V(P)$. If $w$ is the first vertex of $P$, then delete it. Otherwise, the previous vertex $u$ on $P$ has an arc $ur$ to $r$ (the arc $ur$ exists since $A\subseteq V(H_{1})$), and we replace $P$ in $B^{-}_{r}$ by two paths: one is $P[p,u]r$, where $p$ is the first vertex of $P$, and the other is $P[w,r].$

Note that the in-degree $d^{-}(w)$ of $w$ has decreased by one. Thus, after $d^{-}(w)$ such replacements the in-degree of $w$ becomes equal to zero, i.e., $w$ is the first vertex on its maximal path $Q$ to $r$ and therefore $w$ will be deleted when we consider $Q.$ This means that after a finite number of replacements, we will delete all vertices of $A$ in $B^{-}_{r}$ and obtain an in-branching $B_{r}^{\prime-}$ of $Q^{\prime}$ rooted at $r.$ Similarly, we can construct an out-branching $B^{\prime+}_{r}$ of $Q^{\prime}.$ Note that to build $B_{r}^{\prime-}$ we add only arcs to $r$ and to build $B^{\prime+}_{r}$ we add only arcs from $r.$ This fact and the fact that $B^{-}_{r}$ and $B^{+}_{r}$ are arc-disjoint, imply that $B^{\prime-}_{r}$ and $B^{\prime+}_{r}$ are arc-disjoint, too.

$\Box$

By Theorems 3.1 and 3.2, we immediately have the following:

Theorem 3.3

Given a semicomplete composition and a vertex $r$, we can decide in polynomial time whether $D$ has a good pair rooted at $r.$

4 Open Problems and Related Results

Theorem 3.2 provides a characterization for the following problem for semicomplete compositions: given a digraph $D$ and a vertex $r\in V(D)$ decide whether $D$ has a good pair rooted at $r.$ The theorem generalizes a similar characterization by Bang-Jensen and Huang [7] for quasi-transitive digraphs. Strong semicomplete compositions is not the only class of digraphs generalizing strong quasi-transitive digraphs. Other such classes have been studied such as $k$-quasi-transitive digraphs [12] and it would be interesting to see whether a characterization for the problem (or, at least non-trivial sufficient conditions) on $k$-quasi-transitive digraphs can be obtained. As we mentioned above, Bang-Jensen and Huang [9] obtained a characterization for the problem on locally semicomplete digraphs. It would be interesting to see whether a characterization for the problem on in-locally semicomplete digraphs [2, 4] can be obtained.

An out-branching and in-branching $B^{+}_{r}$ and $B^{-}_{r}$ are called $k$-distinct if $B^{+}_{r}$ has at least $k$ arcs not present in $B^{-}_{r}.$ The problem of deciding whether a digraph $D$ has a $k$-distinct pair of out- and in-branchings is NP-complete since it generalizes the good pair problem ($k=|V(D)|-1$). Bang-Jensen and Yeo [10] asked whether the $k$-distinct problem is fixed-parameter tractable when parameterized by $k$, i.e., whether there is an $O(f(k)|V(D)|^{O(1)})$-time algorithm for solving the problem, where $f(k)$ is an arbitrary computable function in $k$ only. Gutin, Reidl and Wahlström [14] answered this open question in affirmative by designing an $O(2^{O(k\log^{2}k)}|V(D)|^{O(1)})$-time algorithm for solving the problem.