Abstract:
The germination responses of Hordeum ulgare seeds to saline stress caused by different salt types was studied. For this, 25 seeds of mentioned species were placed on filter paper in Petri dishes containing distilled water (control), 60, 120, 180, 240, 300, 360 and 420 mM. saline solution of NaCl, CaCl2 an KCl. The results indicated that saline levels effects were significant (P < 0.05) for seed germination percentage, seed germination velocity, mean time to germination, length of the stem and radicle and seed vigour. Seed germination decreased significantly by increasing salinity levels. Also, the results showed that H. vulgare is more tolerant than H. vulgare against salinity in germination stage.

Abstract:
Let $P$ and $S$ be two disjoint sets of $n$ and $m$ points in the plane, respectively. We consider the problem of computing a Steiner tree whose Steiner vertices belong to $S$, in which each point of $P$ is a leaf, and whose longest edge length is minimum. We present an algorithm that computes such a tree in $O((n+m)\log m)$ time, improving the previously best result by a logarithmic factor. We also prove a matching lower bound in the algebraic computation tree model.

Abstract:
Given a set $P$ of $n$ points in the plane, the order-$k$ Gabriel graph on $P$, denoted by $k$-$GG$, has an edge between two points $p$ and $q$ if and only if the closed disk with diameter $pq$ contains at most $k$ points of $P$, excluding $p$ and $q$. We study matching problems in $k$-$GG$ graphs. We show that a Euclidean bottleneck perfect matching of $P$ is contained in $10$-$GG$, but $8$-$GG$ may not have any Euclidean bottleneck perfect matching. In addition we show that $0$-$GG$ has a matching of size at least $\frac{n-1}{4}$ and this bound is tight. We also prove that $1$-$GG$ has a matching of size at least $\frac{2(n-1)}{5}$ and $2$-$GG$ has a perfect matching. Finally we consider the problem of blocking the edges of $k$-$GG$.

Abstract:
We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle $\triangledown$, and there is an edge between two points in $P$ if and only if there is an empty homothet of $\triangledown$ having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely $k$-TD, which contains an edge between two points if the interior of the homothet of $\triangledown$ having the two points on its boundary contains at most $k$ points of $P$. We consider the connectivity, Hamiltonicity and perfect-matching admissibility of $k$-TD. Finally we consider the problem of blocking the edges of $k$-TD.

Abstract:
Let $P$ be a set of $n$ points in general position in the plane. Given a convex geometric shape $S$, a geometric graph $G_S(P)$ on $P$ is defined to have an edge between two points if and only if there exists an empty homothet of $S$ having the two points on its boundary. A matching in $G_S(P)$ is said to be $strong$, if the homothests of $S$ representing the edges of the matching, are pairwise disjoint, i.e., do not share any point in the plane. We consider the problem of computing a strong matching in $G_S(P)$, where $S$ is a diametral-disk, an equilateral-triangle, or a square. We present an algorithm which computes a strong matching in $G_S(P)$; if $S$ is a diametral-disk, then it computes a strong matching of size at least $\lceil \frac{n-1}{17} \rceil$, and if $S$ is an equilateral-triangle, then it computes a strong matching of size at least $\lceil \frac{n-1}{9} \rceil$. If $S$ can be a downward or an upward equilateral-triangle, we compute a strong matching of size at least $\lceil \frac{n-1}{4} \rceil$ in $G_S(P)$. When $S$ is an axis-aligned square we compute a strong matching of size $\lceil \frac{n-1}{4} \rceil$ in $G_S(P)$, which improves the previous lower bound of $\lceil \frac{n}{5} \rceil$.

Abstract:
The Csp3 atom of the chromenyl fused-ring system in the title compound, C14H16O3, deviates by 0.407 (2) from the plane of the other atoms (r.m.s. deviation = 0.041 ). The ethoxy substituent occupies a pseudo-axial position.

Abstract:
The molecule of the title compound, C21H24BrNO5, has a planar furan ring [maximum deviation = 0.025 (3) ]. The carboxymethyl group in the 3-position is nearly coplanar with this ring [dihedral angle = 7.9 (1)°], whereas that in the 4-position is nearly perpendicular to it [dihedral angle = 78.9 (1) ].

Abstract:
Let $R$ and $B$ be two disjoint sets of points in the plane such that $|B|\leqslant |R|$, and no three points of $R\cup B$ are collinear. We show that the geometric complete bipartite graph $K(R,B)$ contains a non-crossing spanning tree whose maximum degree is at most $\max\left\{3, \left\lceil \frac{|R|-1}{|B|}\right\rceil + 1\right\}$; this is the best possible upper bound on the maximum degree. This solves an open problem posed by Abellanas et al. at the Graph Drawing Symposium, 1996.

Abstract:
Given a point set $P$ and a class $\mathcal{C}$ of geometric objects, $G_\mathcal{C}(P)$ is a geometric graph with vertex set $P$ such that any two vertices $p$ and $q$ are adjacent if and only if there is some $C \in \mathcal{C}$ containing both $p$ and $q$ but no other points from $P$. We study $G_{\bigtriangledown}(P)$ graphs where $\bigtriangledown$ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-$\Theta_6$ graphs and TD-Delaunay graphs. The main result in our paper is that for point sets $P$ in general position, $G_{\bigtriangledown}(P)$ always contains a matching of size at least $\lceil\frac{n-2}{3}\rceil$ and this bound cannot be improved above $\lceil\frac{n-1}{3}\rceil$. We also give some structural properties of $G_{\davidsstar}(P)$ graphs, where $\davidsstar$ is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of $G_{\davidsstar}(P)$ is simply a path. Through the equivalence of $G_{\davidsstar}(P)$ graphs with $\Theta_6$ graphs, we also derive that any $\Theta_6$ graph can have at most $5n-11$ edges, for point sets in general position.

Abstract:
Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be packed into any point set $P$. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem.