On The Complexity of Distance-d Independent Set Reconfiguration
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For a fixed positive integer d ≥ 2, a distance-d independent set (DdIS) of a graph is a vertex subset whose distance between any two members is at least d. Imagine that there is a token placed on each member of a DdIS. Two DdISs are adjacent under Token Sliding (𝖳𝖲) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping (𝖳𝖩), the target vertex needs not to be adjacent to the original one. The Distance-d Independent Set Reconfiguration (DdISR) problem under 𝖳𝖲/𝖳𝖩 asks if there is a corresponding sequence of adjacent DdISs that transforms one given DdIS into another. The problem for d = 2, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of DdISR on different graphs under 𝖳𝖲 and 𝖳𝖩 for any fixed d ≥ 3. On chordal graphs, we show that DdISR under 𝖳𝖩 is in 𝙿 when d is even and 𝙿𝚂𝙿𝙰𝙲𝙴-complete when d is odd. On split graphs, there is an interesting complexity dichotomy: DdISR is 𝙿𝚂𝙿𝙰𝙲𝙴-complete for d = 2 but in 𝙿 for d=3 under 𝖳𝖲, while under 𝖳𝖩 it is in 𝙿 for d = 2 but 𝙿𝚂𝙿𝙰𝙲𝙴-complete for d = 3. Additionally, certain well-known hardness results for d = 2 on general graphs, perfect graphs, planar graphs of maximum degree three and bounded bandwidth can be extended for d ≥ 3.
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