Let p and q be positive integers with p/q ≥ 2. The "reconfiguration problem" for circular colourings asks, given two (p, q)-colourings f and g of a graph G, is it possible to transform f into g by changing the colour of one vertex at a time such that every intermediate mapping is a (p, q)-colouring? We show that this problem can be solved in polynomial time for 2 ≤ p/q < 4 and that it is PSPACE-complete for p/q ≥ 4. This generalizes a known dichotomy theorem for reconfiguring classical graph colourings. As an application of the reconfiguration algorithm, we show that graphs with fewer than (k − 1)!/2 cycles of length divisible by k are k-colourable.
This pilot study investigated the perception of the quality of health care received by 55 HIV-positive African Americans. A survey instrument, "The Quality of Care Through the Patient's Eyes"-HIV questionnaire (QUOTE-HIV), developed in the Netherlands, was used to collect quantitative data from the modified QUOTE-HIV. Qualitative data are from 2 focus groups' perception discussions of the applicability of the QUOTE-HIV to HIV-positive African Americans. The study's purpose was to assess the usefulness of the questionnaire to identify patients' perceptions of quality health care. Results indicated that the QUOTE-HIV is a useful tool to assess HIV-positive African Americans' satisfaction and adequately covered all areas of concern discussed by both focus groups as components for measuring perceived quality of care. Most important, the QUOTE-HIV was found to be appropriate for assessing study participants' satisfaction with quality health care and assessing the receipt of quality health care from their respective providers.
Subgraph reconfiguration is a family of problems focusing on the reachability of the solution space in which feasible solutions are subgraphs, represented either as sets of vertices or sets of edges, satisfying a prescribed graph structure property. Although there has been previous work that can be categorized as subgraph reconfiguration, most of the related results appear under the name of the property under consideration; for example, independent set, clique, and matching. In this paper, we systematically clarify the complexity status of subgraph reconfiguration with respect to graph structure properties.
For a fixed graph H, the reconfiguration problem for H-colourings (i.e. homomorphisms to H) asks: given a graph G and two H-colourings ϕ and ψ of G, does there exist a sequence f0, . . . , fm of H-colourings such that f0 = ϕ, fm = ψ and fi(u)fi+1(v) ∈ E(H) for every 0 ≤ i < m and uv ∈ E(G)? If the graph G is loop-free, then this is the equivalent to asking whether it possible to transform ϕ into ψ by changing the colour of one vertex at a time such that all intermediate mappings are H-colourings. In the affirmative, we say that ϕ reconfigures to ψ. Currently, the complexity of deciding whether an H-colouring ϕ reconfigures to an H-colouring ψ is only known when H is a clique, a circular clique, a C4-free graph, or in a few other cases which are easily derived from these. We show that this problem is PSPACE-complete when H is an odd wheel.An important notion in the study of reconfiguration problems for H-colourings is that of a frozen H-colouring; i.e. an H-colouring ϕ such that ϕ does not reconfigure to any Hcolouring ψ such that ψ = ϕ. We obtain an explicit dichotomy theorem for the problem of deciding whether a given graph G admits a frozen H-colouring. The hardness proof involves a reduction from a CSP problem which is shown to be NP-complete by establishing the non-existence of a certain type of polymorphism.
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