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.
Supported in part by grants NSF-INT-9815095/DAAD-315-PPP-g ü-ab, NSF-CCR-0311021, and a DAAD grant.
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Hemaspaandra, E., Spakowski, H., Thakur, M. (2004). Complexity of Cycle Length Modularity Problems in Graphs. In: Farach-Colton, M. (eds) LATIN 2004: Theoretical Informatics. LATIN 2004. Lecture Notes in Computer Science, vol 2976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24698-5_54
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DOI: https://doi.org/10.1007/978-3-540-24698-5_54
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