Graph Isomorphism Restricted by Lists
Pavel Klavik, Du\v{s}an Knop, Peter Zeman

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 and , 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 , we are given a list of possible images of . 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:…
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