Trees with Maximum p-Reinforcement Number

Let $G=(V,E)$ be a graph and $p$ a positive integer. The $p$-domination number \g_p(G) is the minimum cardinality of a set $D\subseteq V$ with $|N_G(x)\cap D|\geq p$ for all $x\in V\setminus D$. The $p$-reinforcement number $r_p(G)$ is the smallest number of edges whose addition to $G$ results in a graph $G'$ with \g_p(G')<\g_p(G). Recently, it was proved by Lu et al. that $r_p(T)\leq p+1$ for a tree $T$ and $p\geq 2$. In this paper, we characterize all trees attaining this upper bound for $p\geq 3$

The p-Domination Number of Complete Multipartite Graphs

Let $G=(V,E)$ be a graph and $p$ a positive integer. A subset $S\subseteq V$ is called a $p$-dominating set of $G$ if every vertex not in $S$ has at least $p$ neighbors in $S$. The $p$-domination number is the minimum cardinality of a $p$-dominating set in $G$. In this paper, we establish an exact formula of the $p$-domination number of all complete multipartite graphs for arbitrary positive integer $p$.Comment: 9 page

The total bondage number of grid graphs

The total domination number of a graph $G$ without isolated vertices is the minimum number of vertices that dominate all vertices in $G$. The total bondage number $b_t(G)$ of $G$ is the minimum number of edges whose removal enlarges the total domination number. This paper considers grid graphs. An $(n,m)$-grid graph $G_{n,m}$ is defined as the cartesian product of two paths $P_n$ and $P_m$. This paper determines the exact values of $b_t(G_{n,2})$ and $b_t(G_{n,3})$, and establishes some upper bounds of $b_t(G_{n,4})$.Comment: 15 pages with 4 figure

Large-scale in silico mapping of complex quantitative traits in inbred mice

Understanding the genetic basis of common disease and disease-related quantitative traits will aid in the development of diagnostics and therapeutics. The processs of gene discovery can be sped up by rapid and effective integration of well-defined mouse genome and phenome data resources. We describe here an in silico gene-discovery strategy through genome-wide association (GWA) scans in inbred mice with a wide range of genetic variation. We identified 937 quantitative trait loci (QTLs) from a survey of 173 mouse phenotypes, which include models of human disease (atherosclerosis, cardiovascular disease, cancer and obesity) as well as behavioral, hematological, immunological, metabolic, and neurological traits. 67% of QTLs were refined into genomic regions <0.5 Mb with ∼40-fold increase in mapping precision as compared with classical linkage analysis. This makes for more efficient identification of the genes that underlie disease. We have identified two QTL genes, Adam12 and Cdh2, as causal genetic variants for atherogenic diet-induced obesity. Our findings demonstrate that GWA analysis in mice has the potential to resolve multiple tightly linked QTLs and achieve single-gene resolution. These high-resolution QTL data can serve as a primary resource for positional cloning and gene identification in the research community

Multi-segment osteotomy with interlocking intramedullary nail fixation in the treatment of lower limb deformity in older children with hypophosphatemic rickets

ObjectiveMalformations of the lower limbs caused by hypophosphatemic rickets in older children are mostly complex, occurring on multiple planes without a single apex and showing arcuate bending of the diaphysis combined with torsion deformity, and are difficult to correct. This study retrospectively investigated the effect of and indicators for multi-segment osteotomy with interlocking intramedullary nail fixation in the treatment of bony deformity caused by hypophosphatemic rickets.MethodsThe clinical data of 21 hypophosphatemic rickets patients seen between August 2007 and March 2022 were collected. The age range of the patients at the first surgery was 11 years and 1 month old to 15 years and 3 months old, with an average age of 12 years and 8 months. There were 6 males and 15 females. All patients had abnormal alignment of their lower limbs, with 32 limbs having varus deformity and 10 limbs having valgus deformity.ResultsA total of 67 surgeries were performed across the 21 patients, including 24 cases of femoral osteotomy with antegrade intramedullary nail fixation, 6 cases of femoral osteotomy with retrograde intramedullary nail fixation, and 20 cases of tibial osteotomy with interlocking intramedullary nail fixation. A total of 34 limbs eventually underwent interlocking intramedullary nail fixation, 9 with genu valgum and 25 with genu varus. All 21 patients were followed up for a period of 14∼96 months, with an average of 42.6 months. The ends of the osteotomies achieved bony union in 4–9 months (average 6.8 months), after which normal weight-bearing walking could be resumed. No infection, vascular or neurological complications, or nonunion occurred. During postoperative follow-up, the alignment the lower limbs passed through zone 1 in 13 limbs, zone 2 in 12 limbs, and zone 3 in 5 limbs. The overall rate of an excellent effect was 83.3%.ConclusionFor lower limb deformity caused by hypophosphatemic rickets in older children, multi-segment osteotomy and strong fixation with interlocking intramedullary nails can achieve good correction outcomes