Graph isomorphism restricted by lists

The complexity of graph isomorphism (GraphIso ) is a famous problem in 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 ∈ V ( G ), we are given a list L ( u ) ⊆ V ( H ) of possible images of u. After 35 years, we revive the study of this problem and consider which results for GraphIso can be modified to solve ListIso. We prove: 1) Under certain conditions, GI -completeness of a class of graphs implies NP -completeness of ListIso. 2) Several combinatorial algorithms for GraphIso can be modified to solve ListIso: for trees, planar graphs, interval graphs, circle graphs, permutation graphs, and bounded treewidth graphs. 3) ListIso is NP -complete for cubic colored graphs with sizes of color classes bounded by 8 with all lists of size at most 3.

Authors

• Pavel Klavík
• Dušan Knop Czech Technical University in Prague
• Peter Zeman

Keywords

• Graph isomorphism problem
• Restricted computational problem
• Polynomial-time algorithms
• NP-completeness
• Bounded treewidth
• Bounded degree graphs