A note on the convexity number for complementary prisms

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$.Comment: 10 pages, 2 figure

A note on the convexity number for complementary prisms

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is $\textit{convex}$ if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The $\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. In this work, we we prove that the decision problem related to the convexity number is NP-complete even restricted to complementary prisms, we determine $con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we present a lower bound when $diam(G) \neq 3$

A polynomial time algorithm for geodetic hull number for complementary prisms

Let G be a finite, simple, and undirected graph and let S ⊆ V (G). In the geodetic convexity, S is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The convex hull H(S) of S is the smallest convex set containing S. The hull number h(G) is the minimum cardinality of a set S ⊆ V (G) such that H(S) = V (G). The complementary prism GG̅ GG̅ of a graph G arises from the disjoint union of the graph G and G̅ G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅ G̅ . Previous works have determined h(GG̅) h(GG̅) when both G and G̅ G̅ are connected and partially when G is disconnected. In this paper, we characterize convex sets in GG̅ GG̅ and we present equalities and tight lower and upper bounds for h(GG̅) h(GG̅) . This fills a gap in the literature and allows us to show that h(GG̅) h(GG̅) can be determined in polynomial time, for any graph G

On the geodetic hull number for complementary prisms II

In the geodetic convexity, a set of vertices S of a graph G is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The convex hull H(S) of S is the smallest convex set containing S. If H(S) = V (G), then S is a hull set. The cardinality h(G) of a minimum hull set of G is the hull number of G. The complementary prism GḠ of a graph G arises from the disjoint union of the graph G and Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A graph G is autoconnected if both G and Ḡ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When G is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when G is a non-split graph, it is limited by 3

Rutin administration attenuates myocardial dysfunction in diabetic rats

Oxidative stress plays a major role in diabetic cardiomyopathy pathogenesis. Anti-oxidant therapy has been investigated in preventing or treating several diabetic complications. However, anti-oxidant action on diabetic-induced cardiac remodeling is not completely clear. This study evaluated the effects of rutin, a flavonoid, on cardiac and myocardial function in diabetic rats. Wistar rats were assigned into control (C, n = 14); control-rutin (C-R, n = 14); diabetes mellitus (DM, n = 16); and DM-rutin (DM-R, n = 16) groups. Seven days after inducing diabetes (streptozotocin, 60 mg/kg, i.p.), rutin was injected intraperitoneally once a week (50 mg/kg) for 7 weeks. Echocardiogram was performed and myocardial function assessed in left ventricular (LV) papillary muscles. Serum insulin concentration was measured by ELISA. One-way ANOVA and Tukey's post hoc test. Glycemia was higher in DM than DM-R and C and in DM-R than C-R. Insulin concentration was lower in diabetic groups than controls (C 2.45 ± 0.67; C-R 2.09 ± 0.52; DM 0.59 ± 0.18; DM-R 0.82 ± 0.21 ng/mL). Echocardiogram showed no differences between C-R and C. DM had increased LV systolic diameter compared to C, and increased left atrium diameter/body weight (BW) ratio and LV mass/BW ratio compared to C and DM-R. Septal wall thickness, LV diastolic diameter/BW ratio, and relative wall thickness were lower in DM-R than DM. Fractional shortening and posterior wall shortening velocity were lower in DM than C and DM-R. In papillary muscle preparation, DM and DM-R presented higher time to peak tension and time from peak tension to 50% relaxation than controls; time to peak tension was lower in DM-R than DM. Under 0.625 and 1.25 mM extracellular calcium concentrations, DM had higher developed tension than C. Rutin attenuates cardiac remodeling and left ventricular and myocardial dysfunction caused by streptozotocin-induced diabetes mellitus.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP