Domination in Functigraphs

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid u \in V(G_1), v \in V(G_2), v=f(u)\}$. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let $\gamma(G)$ denote the domination number of $G$. It is readily seen that $\gamma(G) \le \gamma(C(G,f)) \le 2 \gamma(G)$. We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.Comment: 18 pages, 8 figure

Metric dimension and zero forcing number of two families of line graphs

summary:Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the “AIM Minimum Rank–Special Graphs Work Group”, whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that $Z(G) \le 2Z(L(G))$ for a simple and connected graph $G$. Further, we show that $Z(G) \le Z(L(G))$ when $G$ is a tree or when $G$ contains a Hamiltonian path and has a certain number of edges. We compare the metric dimension with the zero forcing number of a line graph by demonstrating a couple of inequalities between the two parameters. We end by stating some open problems

Measured sodium excretion is associated with cardiovascular outcomes in non-dialysis CKD patients: results from the KNOW-CKD study

BackgroundThere are insufficient studies on the effect of dietary salt intake on cardiovascular (CV) outcomes in chronic kidney disease (CKD) patients, and there is no consensus on the sodium (Na) intake level that increases the risk of CV disease in CKD patients. Therefore, we investigated the association between dietary salt intake and CV outcomes in CKD patients.MethodsIn the Korean cohort study for Outcome in patients with CKD (KNOW-CKD), 1,937 patients were eligible for the study, and their dietary Na intake was estimated using measured 24h urinary Na excretion. The primary outcome was a composite of CV events and/or all-cause death. The secondary outcome was a major adverse cardiac event (MACE).ResultsAmong 1,937 subjects, there were 205 (10.5%) events for the composite outcome and 110 (5.6%) events for MACE. Compared to the reference group (urinary Na excretion< 2.0g/day), the group with the highest measured 24h urinary Na excretion (urinary Na excretion ≥ 8.0g/day) was associated with increased risk of both the composite outcome (hazard ratio 3.29 [95% confidence interval 1.00-10.81]; P = 0.049) and MACE (hazard ratio 6.28 [95% confidence interval 1.45-27.20]; P = 0.013) in a cause-specific hazard model. Subgroup analysis also showed a pronounced association between dietary salt intake and the composite outcome in subgroups of patients with abdominal obesity, female, lower estimated glomerular filtration rate (< 60 ml/min per 1.73m2), no overt proteinuria, or a lower urinary potassium-to-creatinine ratio (< 46 mmol/g).ConclusionA high-salt diet is associated with CV outcomes in non-dialysis CKD patients

On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure