The Isomorphism Problem for Planar 3-Connected Graphs is in Unambiguous Logspace

TL;DR

This paper improves the complexity bounds for the isomorphism problem in planar 3-connected graphs, showing it is solvable in unambiguous logspace, and derives related complexity results for oriented graphs.

Contribution

The paper establishes that the isomorphism problem for planar 3-connected graphs is in unambiguous logspace, a significant complexity class improvement over previous bounds.

Findings

Isomorphism problem for planar 3-connected graphs is in UL ∩ coUL.

The problem for oriented graphs is in NL.

The problems are L-hard, indicating their computational difficulty.

Abstract

The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3-connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class $AC^1$. In this paper we improve the upper bound for planar 3-connected graphs to unambiguous logspace, in fact to $UL \cap coUL$. As a consequence of our method we get that the isomorphism problem for oriented graphs is in $NL$. We also show that the problems are hard for $L$.