Erdős-Pósa property of tripods in directed graphs

TL;DR

This work establishes the Erdős-Pósa property for tripods in directed graphs with designated sources and sinks: either there are $k$ vertex-disjoint tripods or a hitting set of size $f(k)$ meets all tripods. The authors define tripods as unions of two disjoint $S$-$c$ paths and a $c$-$t$ path sharing only the centre, and develop a robust framework combining onion structures, Ramsey-type arguments, and a separation-based Erdős-Pósa strategy. A key contribution is the construction of the hitting function $f(k)$ via $f(k)=2f(k-1)+2g(k)$, where $g(k)$ governs a packing lemma for crossing linkages; matroid intersection tools bound the cut structure, enabling the inductive proof. The paper also links tripods to directed immersions and discusses extensions to edge-disjoint tripods and to acyclic patterns such as $k$-arborescences, highlighting both potential gains and obstacles.

Abstract

Let $D$ be a directed graphs with distinguished sets of sources $S\subseteq V(D)$ and sinks $T\subseteq V(D)$. A tripod in $D$ is a subgraph consisting of the union of two $S$-$T$-paths that have distinct start-vertices and the same end-vertex, and are disjoint apart from sharing a suffix. We prove that tripods in directed graphs exhibit the Erdős-Pósa property. More precisely, there is a function $f\colon \mathbb{N}\to \mathbb{N}$ such that for every digraph $D$ with sources $S$ and sinks $T$, if $D$ does not contain $k$ vertex-disjoint tripods, then there is a set of at most $f(k)$ vertices that meets all the tripods in $D$.

Figures (4)

• Figure 1: A tripod with its centre $c$, the two sources $s_1$ and $s_2$ and its sink $t$.
• Figure 2: On the left an onion and to the right an onion-star of order $3$.
• Figure 3: The two linkages and the new vertex $x$. We also illustrate how an onion with root $x$ contains a subgraph that corresponds to a tripod in the migration digraph $D$.
• Figure 4: How two paths $P_1$ and $P_2$ from $\mathcal{P}( B_i ) - \mathcal{P}( B_j )$ might look and how they yield a tripod $R$.

Theorems & Definitions (26)

• Theorem 1
• Corollary 2
• Theorem 3: Menger's Theorem menger1927directedmenger
• Theorem 4
• Lemma 5: Bożyk, Defrain, Okrasa, Pilipczuk bozyk2022digraphs
• Corollary 6
• proof
• Lemma 7
• Lemma 8
• proof : Proof of \ref{['thm:crossing_linkages']}