Strong chromatic index of sparse graphs

A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $\chi_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In this note we prove that $\chi_{s}^{\prime}(G)\leq (4k-1)\Delta (G)-k(2k+1)+1$ for every $k$-degenerate graph $G$. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that $$ for any chordless graph $G$. Both bounds remain valid for the list version of the strong edge coloring of these graphs

Centroidal localization game

One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph $G$ is the minimum number $k$ of detectors placed in some vertices $\{v_1,\cdots,v_k\}$ such that the vector $(d_1,\cdots,d_k)$ of the distances $d(v_i,r)$ between the detectors and the entity's location $r$ allows to uniquely determine $r \in V(G)$. In a more realistic setting, instead of getting the exact distance information, given devices placed in $\{v_1,\cdots,v_k\}$, we get only relative distances between the entity's location $r$ and the devices (for every $1\leq i,j\leq k$, it is provided whether $d(v_i,r) >$, $<$, or $=$ to $d(v_j,r)$). The centroidal dimension of a graph $G$ is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set $\{v_1,\cdots,v_k\}$ of vertices is probed and then the relative distances between the vertices $v_i$ and the current location $r$ of the entity are given. If not located, the moving entity may move along one edge. Let $\zeta^* (G)$ be the minimum $k$ such that the entity is eventually located, whatever it does, in the graph $G$. We prove that $\zeta^* (T)\leq 2$ for every tree $T$ and give an upper bound on $\zeta^*(G\square H)$ in cartesian product of graphs $G$ and $H$. Our main result is that $\zeta^* (G)\leq 3$ for any outerplanar graph $G$. We then prove that $\zeta^* (G)$ is bounded by the pathwidth of $G$ plus 1 and that the optimization problem of determining $\zeta^* (G)$ is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn

Localization game on geometric and planar graphs

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph $G$ we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant $\zeta (G)$, defined as the least number of cops needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth $2$ and unbounded $\zeta (G)$. On a positive side, we prove that $\zeta (G)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $\zeta (G)$ is NP-hard in graphs with diameter at most $2$. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane

Novel Bradykinin Analogues Modified in the N-Terminal Part of the Molecule with a Variety of Acyl Substituents

In the current work we present some pharmacological characteristics of ten new analogues of bradykinin (Arg–Pro–Pro–Gly–Phe–Ser–Pro–Phe–Arg) modified in the N-terminal part of the molecule with a variety of acyl substituents. Of the many acylating agents used previously with B2 receptor antagonists, the following residues were chosen: 1-adamantaneacetic acid (Aaa), 1-adamantanecarboxylic acid (Aca), 4-tert-butylbenzoic acid (t-Bba), 4-aminobenzoic acid (Aba), 12-aminododecanoic acid (Adc), succinic acid (Sua), 4-hydroxybenzoic acid, 4-hydroxy-3-methoxybenzoic acid, 3-(4-hydroxyphenyl)propionic acid and 6-hydroxy-2-naphthoic acid. Biological activity of the compounds was assessed in the in vivo rat blood pressure test and the in vitro rat uterus test. Surprisingly, N-terminal substitution of the bradykinin peptide chain itself with aforementioned groups resulted in antagonists of bradykinin in the pressor test and suppressed agonistic potency in the uterotonic test. These interesting findings need further studies as they can be helpful for designing more potent B2 receptor blockers

New bradykinin analogues acylated on the N-terminus: effect on rat uterus and blood pressure

Our previous studies suggested that acylation of the N-terminus of several known B2 antagonists with various kinds of bulky acyl groups consistently improved their antagonistic potency in rat blood pressure assay. On the other hand, our earlier observations also seemed to suggest that the effects of acylation on the contractility of isolated rat uterus depended substantially on the chemical character of the acyl group, as we observed that this modification might either change the range of antagonism or even transform it into agonism. Bearing all this in mind, we decided to synthesize seven new analogues of bradykinin by N-terminal acylation with various acyl groups of a moderately potent B2 antagonist, previously synthesized by Stewart's group, D-Arg-Arg-Pro-Hyp-Gly-Thr-Ser-D-Phe-Thi-Arg. The analogues were tested in vitro for their blood pressure-lowering and uterotonic activities. The modifications either preserved or increased the antagonistic potency in the rat blood pressure test. On the other hand, all seven substituents negatively influenced the interaction with the rat uterine receptors. Our results may be helpful for designing new B2 agonists and antagonists

New bradykinin B2 receptor antagonists - influence of C-terminal segment modifications on their pharmacological properties

In the present study we describe the synthesis and some pharmacological properties of eight new analogues of bradykinin (BK). Two peptides were designed by substitution of position 7 or 8 of the known [d-Arg0,Hyp3,Thi5,8,d-Phe7]BK antagonist (Stewart's antagonist) with l-pipecolic acid (l-Pip). The next two analogues were obtained by replacement of the d-Phe residue in position 7 of the Stewart's peptide with l-β2-isoproline (l-β2-iPro) or l-β3-homoproline (l-β3-hPro). The four analogues mentioned above were also prepared in N-acylated form with 1-adamantaneacetic acid (Aaa). Biological activity of the compounds was assessed by isolated rat uterus and rat blood pressure tests. Our results showed that l-Pip in position 7 slightly increased antagonistic potency in the blood pressure test, but it turned the analogue into an agonist in the rat uterus test. Replacement of Thi by l-Pip in position 8 also enhanced antagonism in the rat pressure test but preserved the antagonism in the rat uterus test. l-β2-iPro or l-β3-hPro in position 7 decreased the potencies in both tests. We also demonstrated that acylation of the N-terminus did not increase, as was claimed previously, the antagonistic potencies of the resulting peptides. The results thus support the hypothesis about the existence of different types of BK receptors in the rat uterus and blood vessels. Our studies provide new information about the structure-activity relationship of BK antagonists which may help in designing more potent BK receptor blockers