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- W2012123623 abstract "Let P be a family of dipaths of a DAG (Directed Acyclic Graph) G . The load of an arc is the number of dipaths containing this arc. Let π ( G , P ) be the maximum of the load of all the arcs and let w ( G , P ) be the minimum number of wavelengths (colors) needed to color the family of dipaths P in such a way that two dipaths with the same wavelength are arc-disjoint. There exist DAGs such that the ratio between w ( G , P ) and π ( G , P ) cannot be bounded. An internal cycle is an oriented cycle such that all the vertices have at least one predecessor and one successor in G (said otherwise every cycle contains neither a source nor a sink of G ). We prove that, for any family of dipaths P , w ( G , P ) = π ( G , P ) if and only if G is without internal cycle. We also consider a new class of DAGs, called UPP-DAGs, for which there is at most one dipath from a vertex to another. For these UPP-DAGs we show that the load is equal to the maximum size of a clique of the conflict graph. We prove that the ratio between w ( G , P ) and π ( G , P ) cannot be bounded (a result conjectured in an other article). For that we introduce “good labelings” of the conflict graph associated to G and P , namely labelings of the edges such that for any ordered pair of vertices ( x , y ) there do not exist two paths from x to y with increasing labels." @default.
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- W2012123623 date "2013-09-01" @default.
- W2012123623 modified "2023-10-17" @default.
- W2012123623 title "Directed acyclic graphs with the unique dipath property" @default.
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- W2012123623 doi "https://doi.org/10.1016/j.tcs.2012.06.015" @default.
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