🤖 AI Summary
This study investigates the feasibility and computational complexity of subgraph reconfiguration via local operations while preserving specific graph properties. Introducing buffer space as a resource-augmentation mechanism, the work proposes the first resource-aware reconfiguration model. By leveraging structural parameters such as pathwidth, treewidth, and minor-closed graph classes, combined with complexity analysis and constructive algorithms, it establishes that the reconfiguration problem remains NP-hard both on connected graphs of bounded pathwidth and on general graphs. Furthermore, the study reveals stark differences in buffer requirements across graph classes: planar graphs and those with bounded pathwidth or treewidth necessitate Ω(n) additional buffer vertices, whereas cactus graphs under certain constraints admit reconfiguration with only O(1) buffer space.
📝 Abstract
The subgraph reconfiguration problem asks whether one subgraph can be transformed into another via a sequence of local changes while maintaining a specified graph property. In this work, we focus on the setting where the subgraph is specified by its set of edges. Our contributions in this paper are twofold. First, motivated by the contrast that path reconfiguration is $\textsf{NP}$-hard while tree reconfiguration is solvable in linear time, we prove two generalizations: (1) for any fixed $k$ at least one, reconfiguring connected graphs with pathwidth at most $k$ is $\textsf{NP}$-hard, and (2) for any fixed $k$ at least two, reconfiguring graphs with pathwidth at most $k$ is also $\textsf{NP}$-hard. En route to proving (2), we show a general hardness result that applies to a range of minor-closed graph classes, which we use to show planar graph reconfiguration is also $\textsf{NP}$-hard. Second, given our negative results, we extend the problem to a resource-focused setting, asking how much additional buffer space is needed to turn a non-reconfigurable instance into a reconfigurable one. We show that $Ω(n)$ extra buffer space is needed for planar graphs and graphs with bounded pathwidth and treewidth, while $O(1)$ extra buffer space is sufficient for cactus graphs in a restricted setting.