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The p-competition graph Cp(D) of a digraph D = (V,A) is a graph with V (Cp(D)) = V (D), where an edge between distinct vertices x and y if and only if there exist p distinct vertices v1, v2, ..., vp 2 V such that x → vi, y → vi are arcs of the digraph D for each i = 1, 2, ..., p. In this paper, we obtain that a path with order n ≥ 6 is a p-competition graph if and only if n ≥ p + 3. We also show that for a p-competition graph G with no isolated vertices, G+Kn is a p+n-competition graph. And we obtain that a wheel Wn is a p-competition graph if and only if n ≥ p + 3. [ABSTRACT FROM AUTHOR]

Biodegradable polymers used as vascular stent coatings and stent platforms encounter a major challenge: biocompatibility in vivo, which plays an important role in in-stent restenosis (ISR). Co-formulating amorphous calcium phosphate (ACP) into poly(lactic-co-glycolic acid) (PLGA) or poly-L-lactic acid (PLLA) was investigated to address the issue. For stent coating applications, metal stents were coated with polyethylene-co-vinyl acetate/poly-n-butyl methacrylate (PEVA/PBMA), PLGA or PLGA/ACP composites, and implanted into rat aortas for one and three months. Comparing with both PEVA/PBMA and PLGA groups after one month, the results showed that stents coated with PLGA/ACP had significantly reduced restenosis (PLGA/ACP vs. PEVA/PBMA vs. PLGA: 21.24 +/- 2.59% vs. 27.54 +/- 1.19% vs. 32.12 +/- 3.93%, P < 0.05), reduced inflammation (1.25 +/- 0.35 vs. 1.77 +/- 0.38 vs. 2.30 +/- 0.21, P < 0.05) and increased speed of re-endothelialization (1.78 +/- 0.46 vs. 1.17 +/- 0.18 vs. 1.20 +/- 0.18, P < 0.05). After three months, the PLGA/ACP group still displayed lower inflammation score (1.33 +/- 0.33 vs. 2.27 +/- 0.55, P < 0.05) and higher endothelial scores (2.33 +/- 0.33 vs. 1.20 +/- 0.18, P < 0.05) as compared with the PEVA/PBMA group. Moreover, for stent platform applications, PLLA/ACP stent tube significantly reduced the inflammatory cells infiltration in the vessel walls of rabbit iliac arteries relative to their PLLA cohort (NF-kappaB-positive cells: 23.31 +/- 2.33/mm2 vs. 9.34 +/- 1.35/mm2, P < 0.05). No systemic biochemical or pathological evidence of toxicity was found in either PLGA/ACP or PLLA/ACP. The co-formulation of ACP into PLGA and PLLA resulted in improved biocompatibility without systemic toxicity.