Suppose we have a network that is represented by a graph G. Potentially a fire (or other type of contagion) might erupt at some vertex of G. We are able to respond to this outbreak by establishing a firebreak at k other vertices of G, so that the fire cannot pass through these fortified vertices. The question that now arises is which k vertices will result in the greatest number of vertices being saved from the fire, assuming that the fire will spread to every vertex that is not fully behind the k vertices of the firebreak. This is the essence of the Firebreak decision problem, which is the focus of this paper. We establish that the problem is intractable on the class of split graphs as well as on the class of bipartite graphs, but can be solved in linear time when restricted to graphs having constant‐bounded treewidth, or in polynomial time when restricted to intersection graphs. We also consider some closely related problems.
IntroductionTraditionally, lower-limb prostheses are composed of passive components, which provide a fraction of the push-off power of the natural ankle-foot complex. In individuals with transtibial amputation (TTA), this leads to deviations and compensatory mechanisms. Studies have reported significant unloading of the sound limb and knee joint with a powered prosthetic ankle-foot. However, despite the promising biomechanical evidence on unloading, no study has yet investigated the impact of powered prosthetic ankle-foot on musculoskeletal pain.MethodsA total of 250 individuals fit with a powered prosthetic ankle-foot component were invited to participate in an institutional review board–approved cross-sectional study. Participants completed a survey, which collected typical prosthetic history information as well as Numerical Pain Rating Scales across different body regions, the Socket Comfort Score (SCS), the Activity of Daily Living domain of the Knee Injury and Osteoarthritis Outcome Score (KOOS-ADL), and the Oswestry Disability Index (ODI) for both their current and past prosthetic ankle-foot. The differences between results across the two ankle-feet were evaluated in subgroups dependent on the user's current foot.ResultsA total of 57 individuals met the inclusion criteria after completion of the online survey. Forty-one subjects (71.9%) identified as current powered ankle-foot users. Sixteen subjects (28.1%) reported to have used a powered ankle-foot in the past but have since abandoned it. The current powered ankle-foot users' group saw no significant difference in SCSs. The current passive foot users reported significantly (P = 0.002) better socket comfort for the prosthesis with the passive foot. The original and recall-adjusted median ratings of pain in the group of 41 current powered ankle-foot users showed significantly less pain in all three body segments. In the group of 41 current powered ankle-foot users, both the original and recall-adjusted KOOS-ADL and ODI scores were significantly better for the powered ankle-foot.ConclusionsIndividuals in active daily life with TTA may experience relief of sound knee, amputated side knee, and low-back pain, as well as pain-related restrictions in activities of daily living function with use of a powered ankle-foot mechanism.Clinical RelevanceProviding the right patient with a powered ankle-foot has the potential to decrease the individual's pain. The individual may also have fewer pain-related functional restrictions when attempting to achieve activities of daily living.
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