The Complexity of the Empire Colouring Problem

We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly $r > 1$ countries each. We prove that the problem can be solved in polynomial time using $s$ colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than $s/r$. However, if $s \geq 3$, the problem is NP-hard even if the graph is a forest of paths of arbitrary lengths (for any $r \geq 2$, provided $s < 2r - \sqrt(2r + 1/4+ 3/2)$. Furthermore we obtain a complete characterization of the problem's complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any $r \geq 2$, if $3 \leq s \leq 2r-1$ (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if $s<7$ for $r=2$, and $s < 6r-3$, for $r \geq 3$. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than $6r-3$ colours graphs of thickness $r \geq 3$.Comment: 23 pages, 12 figure

The association of lesion eccentricity with plaque morphology and components in the superficial femoral artery: a high-spatial-resolution, multi-contrast weighted CMR study

<p>Abstract</p> <p>Background</p> <p>Atherosclerotic plaque morphology and components are predictors of subsequent cardiovascular events. However, associations of plaque eccentricity with plaque morphology and plaque composition are unclear. This study investigated associations of plaque eccentricity with plaque components and morphology in the proximal superficial femoral artery using cardiovascular magnetic resonance (CMR).</p> <p>Methods</p> <p>Twenty-eight subjects with an ankle-brachial index less than 1.00 were examined with 1.5T high-spatial-resolution, multi-contrast weighted CMR. One hundred and eighty diseased locations of the proximal superficial femoral artery (about 40 mm) were analyzed. The eccentric lesion was defined as [(Maximum wall thickness- Minimum wall thickness)/Maximum wall thickness] ≥ 0.5. The arterial morphology and plaque components were measured using semi-automatic image analysis software.</p> <p>Results</p> <p>One hundred and fifteen locations were identified as eccentric lesions and sixty-five as concentric lesions. The eccentric lesions had larger wall but similar lumen areas, larger mean and maximum wall thicknesses, and more calcification and lipid rich necrotic core, compared to concentric lesions. For lesions with the same lumen area, the degree of eccentricity was associated with an increased wall area. Eccentricity (dichotomous as eccentric or concentric) was independently correlated with the prevalence of calcification (odds ratio 3.78, 95% CI 1.47-9.70) after adjustment for atherosclerotic risk factors and wall area.</p> <p>Conclusions</p> <p>Plaque eccentricity is associated with preserved lumen size and advanced plaque features such as larger plaque burden, more lipid content, and increased calcification in the superficial femoral artery.</p

Leg strength in peripheral arterial disease: associations with disease severity and lower-extremity performance

AbstractObjectiveThe purpose of this study was to determine relationships between lower-extremity arterial obstruction, leg strength, and lower-extremity functioning.DesignThe study design was cross-sectional. A total of 514 outpatients (269 with ankle-brachial index [ABI] <0.90), aged 55 and older, were identified from three Chicago-area hospitals. Individuals with history of lower-extremity revascularization were excluded.Main outcome measuresStrength in each leg, 6-minute walk, 4-meter walking velocity, accelerometer-measured physical activity, and a summary performance score were measured. The summary performance score is a composite measure of lower-extremity functioning, ranging from 0 to 12 (12 = best). The leg with the lower ABI was defined as the “index” leg, and the leg with higher ABI was defined as the “contralateral” leg.ResultsIndex leg ABI levels were associated linearly and significantly with strength for hip extension (P < .001), hip flexion (P < .001), knee extension (P = .066), and knee flexion (P = .003), adjusting for known and potential confounders. In adjusted analyses, the index ABI was also associated linearly and significantly with strength in the contralateral leg. Adjusting for confounders, including ABI, knee extension strength, was associated independently with functional measures.ConclusionAmong patients without prior leg revascularization, strength in each leg is highly correlated with the lower-leg ABI. Leg strength is associated independently with functional performance. Further study is needed to determine whether lower-extremity resistance training improves functioning in patients with peripheral arterial disease