Introduction:
Menger's theorem states the relationship between vertex cuts and internally disjoint paths in a graph.
Coarse graph theory explores metric properties on a large scale, with applications in geometric group theory.
Metric Menger Problem:
Definition of MM(r, k) involves connecting subsets of a graph with k paths of pairwise distance at least r.
NP-completeness proof for MM(r, k) using reduction from 3SAT.
Theorem A:
MM(r, k) is NP-complete for every r ≥ 3 and k ≥ 2, even on graphs with degree at most four.
Parameterized Complexity:
MM(r, k) is in XP when parameterized by treewidth and maximum degree; simpler for r ≤ 3.
Open Questions:
Difficulty analysis of MM(2, k) and its potential as an NP-intermediate problem.
Efficient solutions for MM(n, k) on planar graphs or minor excluded graphs.
"Given an instance of MM(r, k), the constructed instances of MMp(r, k) have the same input graph as the original problem."
"MM(r, k) lies in XP when parameterised by the treewidth and maximum degree of the input graph."