The parity of the length of paths and cycles is a classical and well-studied topic in graph theory and theoretical computer science. The parity constraints can be extended to the label constraints in a group-labeled graph, which is a directed graph with a group label on each arc. Recently, paths and cycles in group-labeled graphs have been investigated, such as finding non-zero disjoint paths and cycles. In this paper, we present a solution to finding an s–t path in a grouplabeled graph with two labels forbidden. This also leads to an elementary solution to finding a zero path in a Z3-labeled graph, which is the first nontrivial case of finding a zero path. This situation in fact generalizes the 2-disjoint paths problem in undirected graphs, which also motivates us to consider that setting. More precisely, we provide a polynomial-time algorithm for testing whether there are at most two possible labels of s–t paths in a group-labeled graph or not, and finding s–t paths attaining at least three distinct labels if exist. We also give a necessary and sufficient condition for a group-labeled graph to have exactly two possible labels of s–t paths, and our algorithm is based on this characterization.
CITATION STYLE
Kawase, Y., Kobayashi, Y., & Yamaguchi, Y. (2015). Finding a path in group-labeled graphs with two labels forbidden. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9134, pp. 797–809). Springer Verlag. https://doi.org/10.1007/978-3-662-47672-7_65
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