1-String CZ-Representation of Planar Graphs

In this paper, we prove that every planar 4-connected graph has a CZ-representation---a string representation using paths in a rectangular grid that contain at most one vertical segment. Furthermore, two paths representing vertices $u,v$ intersect precisely once whenever there is an edge between $u$ and $v$. The required size of the grid is $n \times 2n$

Generator Matrix Based Search for Extremal Self-Dual Binary Error-Correcting Codes

Self-dual doubly even linear binary error-correcting codes, often referred to as Type II codes, are codes closely related to many combinatorial structures such as 5-designs. Extremal codes are codes that have the largest possible minimum distance for a given length and dimension. The existence of an extremal (72,36,16) Type II code is still open. Previous results show that the automorphism group of a putative code C with the aforementioned properties has order 5 or dividing 24. In this work, we present a method and the results of an exhaustive search showing that such a code C cannot admit an automorphism group Z6. In addition, we present so far unpublished construction of the extended Golay code by P. Becker. We generalize the notion and provide example of another Type II code that can be obtained in this fashion. Consequently, we relate Becker's construction to the construction of binary Type II codes from codes over GF(2^r) via the Gray map

11-String B2B_2-VPG Representation of Planar Graphs

In this paper, we prove that every planar graph has a 1-string $B_2$-VPG representation---a string representation using paths in a rectangular grid that contain at most two bends. Furthermore, two paths representing vertices $u,v$ intersect precisely once whenever there is an edge between $u$ and $v$.Comment: arXiv admin note: text overlap with arXiv:1409.581

Improved Bounds for Drawing Trees on Fixed Points with L-shaped Edges

Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$ points in the plane with no two points on the same horizontal or vertical line. It is an open question whether $T$ always has a planar drawing on $P$ such that each edge is drawn as an orthogonal path with one bend (an "L-shaped" edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set $P$ for which such drawings are possible to: $O(n^{1.55})$ for maximum degree 4 trees; $O(n^{1.22})$ for maximum degree 3 (binary) trees; and $O(n^{1.142})$ for perfect binary trees. Drawing ordered trees with L-shaped edges is harder---we give an example that cannot be done and a bound of $O(n \log n)$ points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Restricted String Representations

A string representation of a graph assigns to every vertex a curve in the plane so that two curves intersect if and only if the represented vertices are adjacent. This work investigates string representations of graphs with an emphasis on the shapes of curves and the way they intersect. We strengthen some previously known results and show that every planar graph has string representations where every curve consists of axis-parallel line segments with at most two bends (those are the so-called $B_2$-VPG representations) and simultaneously two curves intersect each other at most once (those are the so-called 1-string representations). Thus, planar graphs are $B_2$-VPG $1$-string graphs. We further show that with some restrictions on the shapes of the curves, string representations can be used to produce approximation algorithms for several hard problems. The $B_2$-VPG representations of planar graphs satisfy these restrictions. We attempt to further restrict the number of bends in VPG representations for subclasses of planar graphs, and investigate $B_1$-VPG representations. We propose new classes of string representations for planar graphs that we call ``order-preserving.'' Order-preservation is an interesting property which relates the string representation to the planar embedding of the graph, and we believe that it might prove useful when constructing string representations. Finally, we extend our investigation of string representations to string representations that require some curves to intersect multiple times. We show that there are outer-string graphs that require an exponential number of crossings in their outer-string representations. Our construction also proves that 1-planar graphs, i.e., graphs that are no longer planar, yet fairly close to planar graphs, may have string representations, but they are not always 1-string

The Proposal of Improving of HR Management of Diverse Teams in High-tech Firm

Diplomová práce se zabývá analýzou personálního řízení ve vybraném high-tech podniku. Teoretická část práce popisuje základní pojmy z oblasti personálního řízení, diverzitních týmů a charakteristiky high-tech sektoru. V praktické části je pak analyzován současný stav pomocí dotazníkového šetření a následně jsou navržena doporučení včetně jejich ekonomického zhodnocení.Master’s thesis deals with the analysis of personnel management in selected high-tech company. The theoretical part of thesis describes the basic concepts of personnel management, diverse teams and characteristics of the high-tech sector. In the practical part is analyzed the current state using questionnaires and subsequently proposed recommendations, including their economic evaluation.

On the Size of Outer-String Representations

Outer-string graphs, i.e., graphs that can be represented as intersection of curves in 2D, all of which end in the outer-face, have recently received much interest, especially since it was shown that the independent set problem can be solved efficiently in such graphs. However, the run-time for the independent set problem depends on N, the number of segments in an outer-string representation, rather than the number n of vertices of the graph. In this paper, we argue that for some outer-string graphs, N must be exponential in n. We also study some special string graphs, viz. monotone string graphs, and argue that for them N can be assumed to be polynomial in n. Finally we give an algorithm for independent set in so-called strip-grounded monotone outer-string graphs that is polynomial in n