Abstract
The parameterized Synchronizing-Road-Coloring Problem (in short: SRCPlC) in its decision version can be formulated as follows: given a digraph G with constant out-degree l, check if G can be synchronized by some word of length C for some synchronizing labeling. We consider the family {SRCPlC}C,l of problems parameterized with constants C and l and try to find for which C and l SRCPlC is NP-complete. It is known that SRCP3C is NP-complete for C≥8. We improve this result by showing that it is so for C≥4 and for l≥3. We also show that SRCPlC is in P for C≤2 and l≥1. Hence, we solve SRCP almost completely for alphabet with 3 or more letters. The case C=3 is still an open problem.
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