On the Complexity of Problems on Graphs Defined on Groups

TL;DR

This paper investigates the computational complexity of various graph problems on group-defined graphs, establishing complexity bounds under ETH and identifying cases where problems are NP-complete or solvable efficiently.

Contribution

It provides new complexity results for graph problems on power and related graphs, including NP-completeness of Weighted Max-Cut and polynomial-time recognition algorithms for certain groups.

Findings

Weighted Max-Cut is NP-complete on power graphs.

Graph Motif cannot be solved in quasipolynomial time on power graphs unless ETH fails.

Recognition of power graphs is polynomial-time for abelian and some nilpotent groups.

Abstract

We study the complexity of graph problems on graphs defined on groups, especially power graphs. We observe that an isomorphism invariant problem, such as Hamiltonian Path, Partition into Cliques, Feedback Vertex Set, Subgraph Isomorphism, cannot be NP-complete for power graphs, commuting graphs, enhanced power graphs, directed power graphs, and bounded-degree Cayley graphs, assuming the Exponential Time Hypothesis (ETH). An analogous result holds for isomorphism invariant group problems: no such problem can be NP-complete unless ETH is false. We show that the Weighted Max-Cut problem is NP-complete in power graphs. We also show that, unless ETH is false, the Graph Motif problem cannot be solved in quasipolynomial time on power graphs, even for power graphs of cyclic groups. We study the recognition problem of power graphs when the adjacency matrix or list is given as input and show that for abelian groups and some classes of nilpotent groups, it is solvable in polynomial time.