The graph isomorphism problem

Abstract

We give an overview of recent advances on the graph isomorphism problem. Our main focus will be on Babai's quasi-polynomial time isomorphism test and subsequent developments that led to the design of isomorphism algorithms with a quasipolynomial parameterized running time of the from n polylog(k) , where k is a graph parameter such as the maximum degree. A second focus will be the combinatorial Weisfeiler-Leman algorithm.

Modularity and Dynamics on Complex Networks

“…Stewart, Golubitsky, and Pivato (2003). For more thorough discussions on the relations between EPs, EEPs, automorphic equivalence and graph isomorphism, we refer to (Chan & Godsil, 1997;Grohe, Kersting, Mladenov, & Selman, 2014;Grohe & Schweitzer, 2020). Important references on how to use these concepts in the context of the analysis of networks and network dynamics include (Cardoso, Delorme, & Rama, 2007;Egerstedt, Martini, Cao, Camlibel, & Bicchi, 2012;Pecora, Sorrentino, Hagerstrom, Murphy, & Roy, 2014;Sanchez-Garcia, 2018).…”

Bayesian probabilistic assignment of chemical shifts in organic solids

“…A given topological atomic environment can be searched by identifying which graphs in the database match the graph of the selected atomic environment. However, there is no known algorithm able to solve the graph isomorphism problem required for each database entry in polynomial time ( 47 , 48 ). Thus, the search was simplified by using the Weisfeiler-Lehman hash ( 49 ) as a unique graph identifier.…”

Trust, transforms, and control flow: A graph-theoretic method to verifying source and binary control flow equivalence

“…Graph Isomorphism (GI) as a computational problem first appears in the chemistry literature of the 1950s as the problem of matching a molecular graph. GI has emerged as one of the few natural problems in the complexity class NP that could neither be classified as being hard (NP-complete) nor shown to be solvable in polynomial-time [29]. Computational Complexity: GI has many application in biology, chemistry, physics, mathematics, and computer science.…”

“…Group Theoretic Algorithms: As the GI and the automorphism group problem are polynomially equivalent, it suffices to solve the latter [44]. These algorithms are used to establish polynomial time bounds for GI problems for graphs of bounded degree [29].…”

“…The isomorphism maps a vertex of the transformed source CFG to a linear block of the binary CFG. Our isomorphism algorithm is based on Weisfeiler-Leman algorithm [50] and color refinement [29,30,31,41]. The vertex coloring uses the distinguished features incorporated in forming the feature vectors.…”