NoteArc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs
Introduction
We use a standard digraph terminology and notation as in [4], [5]. A digraph is strongly connected (or, just strong) if there exists a path from to and a path from to in for every pair of distinct vertices of . An out-tree (in-tree, respectively) rooted at a vertex is an orientation of a tree such that the in-degree (out-degree, respectively) of every vertex but equals one. An out-branching (in-branching, respectively) in a digraph is a spanning subgraph of 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 if and only if has a unique initial strong connectivity component (terminal strong connectivity component, respectively) and 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 [10] characterized digraphs with arc-disjoint out-branchings rooted at a specified vertex . Furthermore, there exists a polynomial algorithm for finding arc-disjoint out-branchings with a given root if they exist [4]. However, it is NP-complete to decide whether a digraph has a pair of arc-disjoint out-branching and in-branching rooted at , which was proved by Thomassen, see [1]. Following [8] we will call such a pair a good pair rooted at.Note that a good pair forms a strong spanning subgraph of and thus if has a good pair, then 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 [8] for locally semicomplete digraphs. Also, Bang-Jensen and Huang [7] showed that if is adjacent to every vertex of (apart from itself) then has a good pair rooted at .
In this paper, we study the existence of good pairs for digraph compositions. Let be a digraph with vertices and let be digraphs such that has vertices . Then the composition is a digraph with vertex set and arc set If is strongly connected, then is called a strong composition and if is semicomplete, i.e., there is at least one arc between every pair of vertices, then is called a semicomplete composition.
Digraph compositions generalize some families of digraphs. In particular, semicomplete compositions generalize strong quasi-transitive digraphs as every strong quasi-transitive digraph is a strong semicomplete composition in which 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 [12], but NP-complete for strong compositions (see, e.g., [6]). When is the same digraph for every , is the lexicographic product of and , see, e.g., [14]. While digraph compositions have been used since 1990s to study quasi-transitive digraphs and their generalizations, see, e.g., [3], [4], [11], the study of digraph decompositions in their own right was initiated only recently in [15].
In the next section, we obtain the following somewhat surprising result: every strong digraph composition in which for every , has a good pair rooted at every vertex of . The condition of 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 and be integers. Then if and , otherwise. In particular, if , will be a shorthand for .
Section snippets
Compositions of digraphs: arbitrary
A digraph has a strong arc decomposition if has two disjoint sets and such that both and are strong. Sun et al. [15] obtained sufficient conditions for a digraph composition to have a strong arc decomposition. In particular, they proved the following:
Theorem 2.1 Let be a digraph with vertices () and let be digraphs. Then has a strong arc decomposition if has a Hamiltonian cycle and one of the following conditions holds: is even and for every
Compositions of digraphs: semicomplete
We use (, respectively) to denote the set of all in-neighbours (out-neighbours, respectively) of a vertex in a digraph .
The next result was obtained by Bang-Jensen and Huang [7].
Theorem 3.1 Let be a strong digraph and a vertex of such that. There is a polynomial-time algorithm to decide whether has a good pair at .
For a path and , let . We now prove the following result on semicomplete compositions which generalizes a similar
Open problems and related results
Theorem 3.2 provides a characterization for the following problem for semicomplete compositions: given a digraph and a vertex decide whether has a good pair rooted at . The theorem generalizes a similar characterization by Bang-Jensen and Huang [7] for quasi-transitive digraphs. Strong semicomplete compositions are not the only class of digraphs generalizing strong quasi-transitive digraphs. Other such classes have been studied such as -quasi-transitive digraphs [11] and it would
Declaration of Competing Interest
The authors declare that there is no conflict of interests regarding the publication of this article.
Acknowledgement
We are very thankful to the reviewer for the suggestions, which improved the presentation.
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This author was supported by Zhejiang Provincial Natural Science Foundation (No. LY20A010013) and National Natural Science Foundation of China (No. 11401389).