Complexity of cycle length modularity problems in graphs

Abstract

The even cycle problem for both undirected [Tho88] and directed [RST99] graphs has been the topic of intense research in the last decade. In this paper, we study the computational complexity of cycle length modularity problems. Roughly speaking, in a cycle length modularity problem, given an input (undirected or directed) graph, one has to determine whether the graph has a cycle C of a specific length (or one of several different lengths), modulo a fixed integer. We denote the two families (one for undirected graphs and one for directed graphs) of problems by (S, m)-UC and (S,m)-DC, where m ∈ ℕ and S ⊆{0, 1,..., m - 1}. (S, m)-UC (respectively, (S, m)-DC) is defined as follows: Given an undirected (respectively, directed) graph G, is there a cycle in G whose length, modulo m, is a member of S? In this paper, we fully classify (i.e., as either polynomial-time solvable or as NP-complete) each problem (S, m)-UC such that 0 ∈ S and each problem (S,m)-DC such that 0∉S. We also give a sufficient condition on S and m for the following problem to be polynomial-time computable: (S, m)-UC such that 0 ∉ S. © Springer-Verlag 2004.