Parameterized aspects of triangle enumeration

Abstract

Listing all triangles in an undirected graph is a fundamental graph primitive with numerous applications. It is trivially solvable in time cubic in the number of vertices. It has seen a significant body of work contributing to both theoretical aspects (e.g., lower and upper bounds on running time, adaption to new computational models) as well as practical aspects (e.g. algorithms tuned for large graphs). Motivated by the fact that the worst-case running time is cubic, we perform a systematic parameterized complexity study of triangle enumeration, providing both positive results (new enumerative kernelizations, “subcubic” parameterized solving algorithms) as well as negative results (uselessness in terms of possibility of “faster” parameterized algorithms of certain parameters such as diameter).