A Polynomial Time Algorithm for Steiner Tree when Terminals Avoid a $K_4$-Minor

TL;DR

This paper presents a polynomial-time algorithm for the Steiner Tree problem in graphs that exclude a specific $K_4$-minor with terminals, extending previous results on planar graphs.

Contribution

It introduces an $O(n^4)$ algorithm for Steiner Tree when terminals avoid a $K_4$-minor, generalizing prior work on planar graphs.

Findings

The problem can be solved in polynomial time under the given minor exclusion.

The algorithm generalizes previous results on planar graphs.

Provides a new approach for Steiner Tree in minor-closed graph classes.

Abstract

We study a special case of the Steiner Tree problem in which the input graph does not have a minor model of a complete graph on 4 vertices for which all branch sets contain a terminal. We show that this problem can be solved in $O(n^4)$ time, where $n$ denotes the number of vertices in the input graph. This generalizes a seminal paper by Erickson et al. [Math. Oper. Res., 1987] that solves Steiner tree on planar graphs with all terminals on one face in polynomial time.