A Menger-type theorem for two induced paths

TL;DR

This work proves an approximate Menger-type theorem for two $X$-$Y$ paths whose union is induced: for any graph $G$, sets $X,Y\subseteq V(G)$, and distance parameter $d\ge1$, either there exist two disjoint $X$-$Y$ paths $P_1,P_2$ with $\mathrm{dist}_G(P_1,P_2)\ge d$ or a vertex $z$ with $B_G(z, c d)$ meeting every $X$-$Y$ path, where $c$ is explicitly bounded by $129$. The proof combines a careful decomposition around a shortest $X$-$Y$ path, a combinatorial analysis of interlaced interval systems, and a construction named a fruit tree with an orchard to produce two distant $X$-$Y$ walks. The method yields a constructive dichotomy and extends to a distance-parametrized version; the authors acknowledge related independent work and discuss potential generalizations to more than two paths. The result advances understanding of how to guarantee separation of $X$-$Y$ paths via either packing or localized hitting sets in finite graphs.

Abstract

We give an approximate Menger-type theorem for when a graph $G$ contains two $X-Y$ paths $P_1$ and $P_2$ such that $P_1 \cup P_2$ is an induced subgraph of $G$. More generally, we prove that there exists a function $f(d) \in O(d)$, such that for every graph $G$ and $X,Y \subseteq V(G)$, either there exist two $X-Y$ paths $P_1$ and $P_2$ such that the distance between $P_1$ and $P_2$ is at least $d$, or there exists $v \in V(G)$ such that the ball of radius $f(d)$ centered at $v$ intersects every $X-Y$ path.

Figures (10)

• Figure 1: A graph without two disjoint anti-complete $\{x_1, x_2, x_3\}-\{y_1, y_2, y_3\}$ paths and no ball of radius one hitting all $X-Y$ paths.
• Figure 2: A clean $(A,B,r)$-system $([a_i,b_i])_{i \in [4]}$ indicated in teal which is interlaced with buffer $\ell$.
• Figure 3: The $(A,B,r)$-system indicated in teal is clean. However, adding the red interval gives an $(A,B,r)$-system which is not clean.
• Figure 4: The components $H_i$ along with their respective $a(H_i)$ and $b(H_i)$ represented by the corresponding vertex in $P$ or a vertex in $X$ or $Y$.
• Figure 5: The fruit tree $\mathcal{W}_i$ of the orchard with its corresponding components (orange) and its composite paths (magenta).

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Conjecture 3
• Theorem 4: McCarty and Seymour, 2023
• proof
• Lemma 5
• proof
• Definition
• Lemma 6
• proof