Given a convex polygon with n vertices in the plane, we are interested in triangulations of its interior, i.e., maximal sets of nonintersecting diagonals that subdivide the interior of the polygon into triangles. The MaxMin area triangulation is the triangulation of the polygon that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. We present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in O(n 2 log n) time and O(n 2 ) space. The algorithms use dynamic programming and a number of geometric properties that are established within the paper.

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The Atom Bond Connectivity index, also known as ABC index was defined by Estrada[4] with relation to the energy of formation of alkanes. It was quickly recognized that this index reflects important structural properties of graphs in general. The ABC index was extensively studied in the last three years, from the point of view of chemical graph theory[5,6], and in general graphs[1]. It was also compared to other structural indices of graphs[2]. Das derives multiple results with implications to the minimum/maximum ABC index on graphs. With relation to trees, it is known that among all the trees of the same number of vertices, the maximum ABC index is attained for the star graph. However, it is not known which tree(s) minimize(s) the ABC index. The problem seems to be hard. It is partially addressed in many sources[5,1,6], but remains open. In this paper we further investigate the trees that minimize the ABC index. Our investigations are limited to chemical trees, i.e. trees in which the maximum vertex degree is 4. The chemical trees were introduced to reflect the structure of the carbon chains and the molecules based on them. Our approach is algorithmic. We identify certain types of edges (chemical bonds) that are important and occur frequently in chemical trees. Further, we study how the removal of a certain edge, the introduction of certain edge or the contraction of certain edge affects the ABC-index of the tree. We pay particular attention to the examples of minimal ABC index chemical trees provided by Dimitrov[3].

We study sets of points in the two-dimensional Euclidean plane. The relative neighbourhood graph (RNG) of a point set is a straight line graph that connects two points from the point set if and only if there is no other point in the set that is closer to both points than they are to each other. A triangulation of a point set is a maximal set of nonintersecting line segments (called edges) with vertices in the point set. We introduce angular rectrictions in the triangulations. Using the well-known method of exclusion regions, we show that the relative neighbourhood graph is a part of every triangulation all of the angles of which are greater than or equal to 30• .

We consider the problem of finding the staircase kernel in orthogonal polygons, with or without holes, in the plane. Orthogonal polygon is a simple polygon in the plane whose sides are either horizontal or vertical. We generalize the notion of visibility in the following way: We say that two points a and b in an orthogonal polygon P are visible to each other via staircase paths if and only if there exist an orthogonal chain connecting a and b and lying entirely in the interior of P. Furthermore, the orthogonal chain should have the property that the angles between the consecutive segments in the chain are either +90∘ or −90 ∘ , and these should alternate along the chain. There are two principal types of staircases, NW-SE and NE-SW. The notion of staircase visibility has been studied in the literature for the last three decades. Based on this notion we can generalize the notion of star-shapedness. A polygon P is called star-shaped under staircase visibility, or simply s-star if and only if there is nonempty set of points S in the interior of P, such that any point of S sees any point of P via staircase path. The largest such set of points is called the staircase kernel of P and denoted ker P. Our work is motivated by the work of Breen [1]. She proves that the staircase kernel of an orthogonal polygon without holes is the intersection of all maximal orthogonally convex polygons contained in it. We extend Breen's results for the case when the orthogonal polygon has holes. We prove the necessary geometric properties, and use them to derive a quadratic time, O( 2 ) algorithm for computing the staircase kernel of an orthogonal polygon with holes, having n vertices in total, including the holes' vertices. The algorithm is based on the plane sweep technique, widely used in Computational Geometry [4]. Our result is optimal in the case of orthogonal polygon with holes, since the kernel (as proven) can consist of quadratic number of disjoint regions. In the case of polygon without holes, there is a linear time algorithm by Gewali[3], that is specific to the case of a polygon without holes. We present examples of our algorithm's results.