Complete Cycle Embedding in Crossed Cubes with Two-Disjoint-Cycle-Cover Pancyclicity

Author(s):  
Tzu-Liang KUNG ◽  
Hon-Chan CHEN
2020 ◽  
Vol 381 ◽  
pp. 125305
Author(s):  
Chao Wei ◽  
Rong-Xia Hao ◽  
Jou-Ming Chang
Keyword(s):  

Author(s):  
Tzu-Liang Kung ◽  
Hon-Chan Chen ◽  
Chia-Hui Lin ◽  
Lih-Hsing Hsu

Abstract A graph $G=(V,E)$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if for any integer $l$ satisfying $r_1 \leq l \leq r_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in $G$ such that the lengths of $C_1$ and $C_2$ are $l$ and $|V(G)| - l$, respectively, where $|V(G)|$ denotes the total number of vertices in $G$. On the basis of this definition, we further propose Ore-type conditions for graphs to be two-disjoint-cycle-cover vertex/edge $[r_1,r_2]$-pancyclic. In addition, we study cycle embedding in the $n$-dimensional locally twisted cube $LTQ_n$ under the consideration of two-disjoint-cycle-cover vertex/edge pancyclicity.


2021 ◽  
Vol 19 (1) ◽  
Author(s):  
Klaus F. Bühler ◽  
Robert Fischer ◽  
Patrice Verpillat ◽  
Arthur Allignol ◽  
Sandra Guedes ◽  
...  

Abstract Background This study compared the effectiveness of recombinant human follicle-stimulating hormone alfa (r-hFSH-alfa; GONAL-f®) with urinary highly purified human menopausal gonadotropin (hMG HP; Menogon HP®), during assisted reproductive technology (ART) treatments in Germany. Methods Data were collected from 71 German fertility centres between 01 January 2007 and 31 December 2012, for women undergoing a first stimulation cycle of ART treatment with r-hFSH-alfa or hMG HP. Primary outcomes were live birth, ongoing pregnancy and clinical pregnancy, based on cumulative data (fresh and frozen-thawed embryo transfers), analysed per patient (pP), per complete cycle (pCC) and per first complete cycle (pFC). Secondary outcomes were pregnancy loss (analysed per clinical pregnancy), cancelled cycles (analysed pCC), total drug usage per oocyte retrieved and time-to-live birth (TTLB; per calendar week and per cycle). Results Twenty-eight thousand six hundred forty-one women initiated a first treatment cycle (r-hFSH-alfa: 17,725 [61.9%]; hMG HP: 10,916 [38.1%]). After adjustment for confounding variables, treatment with r-hFSH-alfa versus hMG HP was associated with a significantly higher probability of live birth (hazard ratio [HR]-pP [95% confidence interval (CI)]: 1.10 [1.04, 1.16]; HR-pCC [95% CI]: 1.13 [1.08, 1.19]; relative risk [RR]-pFC [95% CI]: 1.09 [1.05, 1.15], ongoing pregnancy (HR-pP [95% CI]: 1.10 [1.04, 1.16]; HR-pCC [95% CI]: 1.13 [1.08, 1.19]; RR-pFC [95% CI]: 1.10 [1.05, 1.15]) and clinical pregnancy (HR-pP [95% CI]: 1.10 [1.05, 1.14]; HR-pCC [95% CI]: 1.14 [1.10, 1.19]; RR-pFC [95% CI]: 1.10 [1.06, 1.14]). Women treated with r-hFSH-alfa versus hMG HP had no statistically significant difference in pregnancy loss (HR [95% CI]: 1.07 [0.98, 1.17], were less likely to have a cycle cancellation (HR [95% CI]: 0.91 [0.84, 0.99]) and had no statistically significant difference in TTLB when measured in weeks (HR [95% CI]: 1.02 [0.97, 1.07]; p = 0.548); however, r-hFSH-alfa was associated with a significantly shorter TTLB when measured in cycles versus hMG HP (HR [95% CI]: 1.07 [1.02, 1.13]; p = 0.003). There was an average of 47% less drug used per oocyte retrieved with r-hFSH-alfa versus hMG HP. Conclusions This large (> 28,000 women), real-world study demonstrated significantly higher rates of cumulative live birth, cumulative ongoing pregnancy and cumulative clinical pregnancy with r-hFSH-alfa versus hMG HP.


Author(s):  
Vera Traub ◽  
Thorben Tröbst

AbstractWe consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing problem (CVRP) and other cycle cover problems such as min-max cycle cover and bounded cycle cover. We show that a greedy algorithm followed by a post-processing step yields a $$(2 + \frac{2}{7})$$ ( 2 + 2 7 ) -approximation for this problem by comparing the solution to a polymatroid relaxation. We also show that the analysis of our algorithm is tight and provide a $$2 + \epsilon $$ 2 + ϵ lower bound for the relaxation.


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