Graph Isomorphism Restricted by Lists

TL;DR

This paper investigates list restricted graph isomorphism (ListIso), showing how complexity and algorithms for standard graph isomorphism translate to ListIso, revealing NP-completeness in certain cases and limitations of group-theoretic approaches.

Contribution

It establishes the complexity relationship between GraphIso and ListIso, demonstrating which algorithms translate and identifying NP-hard cases for ListIso.

Findings

GraphIso complexity translates to ListIso NP-completeness in certain classes.

Combinatorial algorithms for GraphIso extend to ListIso for various graph classes.

Group-theoretic algorithms do not translate, leaving ListIso NP-complete for cubic graphs.

Abstract

The complexity of graph isomorphism (GraphIso) is a famous unresolved problem in theoretical computer science. For graphs $G$ and $H$, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list restricted graph isomorphism (ListIso) is NP-complete: for each $u \in V(G)$, we are given a list ${\mathfrak L}(u) \subseteq V(H)$ of possible images of $u$. After 35 years, we revive the study of this problem and consider which results for GraphIso translate to ListIso. We prove the following: 1) When GraphIso is GI-complete for a class of graphs, it translates into NP-completeness of ListIso. 2) Combinatorial algorithms for GraphIso translate into algorithms for ListIso: for trees, planar graphs, interval graphs, circle graphs, permutation graphs, bounded genus graphs, and bounded treewidth graphs. 3) Algorithms based on group theory do not translate: ListIso remains NP-complete for cubic colored graphs with sizes of color classes bounded by 8. Also, ListIso allows to classify results for the graph isomorphism problem. Some algorithms are robust and translate to ListIso. A fundamental problem is to construct a combinatorial polynomial-time algorithm for cubic graph isomorphism, avoiding group theory. By the 3rd result, ListIso is NP-hard for them, so no robust algorithm for cubic graph isomorphism exists, unless P = NP.