On the Simultaneous Minimum Spanning Trees Problem

TL;DR

This paper explores the problem of finding minimum spanning trees across multiple graphs with shared edges, revealing polynomial solutions for unweighted cases and NP-completeness for weighted cases when three or more graphs are involved.

Contribution

It introduces the simultaneous minimum spanning trees problem, analyzing its computational complexity and establishing polynomial solvability for unweighted cases and NP-completeness for weighted cases with three or more graphs.

Findings

Unweighted case is polynomial-time solvable.

Weighted case is polynomial-time solvable for two graphs.

Weighted case is NP-complete for three or more graphs.

Abstract

Simultaneous Embedding with Fixed Edges (SEFE) is a problem where given $k$ planar graphs we ask whether they can be simultaneously embedded so that the embedding of each graph is planar and common edges are drawn the same. Problems of SEFE type have inspired questions of Simultaneous Geometrical Representations and further derivations. Based on this motivation we investigate the generalization of the simultaneous paradigm on the classical combinatorial problem of minimum spanning trees. Given $k$ graphs with weighted edges, such that they have a common intersection, are there minimum spanning trees of the respective graphs such that they agree on the intersection? We show that the unweighted case is polynomial-time solvable while the weighted case is only polynomial-time solvable for $k=2$ and it is NP-complete for $k\geq 3$.