Arc-disjoint in- and out-branchings in semicomplete split digraphs

2024/10/16 by Jiangdong Ai, Ai, Jiangdong, Yiming Hao +5
Computer Science · #Advanced Algebra and Logic #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2410.12575

openalex publication_date 2024/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An out-tree (in-tree) is an oriented tree where every vertex except one, called the root, has in-degree (out-degree) one. An out-branching B⁺_u (in-branching B^-_u) of a digraph D is a spanning out-tree (in-tree) rooted at u. A good (u,v)-pair in D is a pair of branchings B⁺_u, B^-_v which are arc-disjoint. Thomassen proved that deciding whether a digraph has any good pair is NP-complete. A semicomplete split digraph is a digraph where the vertex set is the disjoint union of two non-empty sets, V₁ and V₂, such that V₁ is an independent set, the subdigraph induced by V₂ is semicomplete, and every vertex in V₁ is adjacent to every vertex in V₂. In this paper, we prove that every 2-arc-strong semicomplete split digraph D contains a good (u, v)-pair for any choice of vertices u, v of D, thereby confirming a conjecture by Bang-Jensen and Wang [Bang-Jensen and Wang, J. Graph Theory, 2024].

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