The induced two paths problem

05/08/2023

∙

by   Sandra Albrechtsen, et al.

∙

0

∙

share

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 ∪ P_2 is an induced subgraph of G. More generally, we prove that there exists a function f(d) ∈ O(d), such that for every graph G and X,Y ⊆ 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 ∈ V(G) such that the ball of radius f(d) centered at v intersects every X-Y path.