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Abstract: Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number. We show that the page number of an ordered graph with $n$ vertices and $m$ edges can be computed in $2^m \cdot n^{{\mathcal O}(1)}$ time. An ${\mathcal O}(\log n)$-approximation of this number can be computed efficiently. We can decide in $2^{{\mathcal O}(d \sqrt{k} \log (d+k))} \cdot n^{{\mathcal O}(1)}$ time whether it suffices to delete $k$ edges of an ordered graph to obtain a $d$-planar layout (where every edge crosses at most $d$ other edges) on one page. As an additional parameter, we consider the size $h$ of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For $h = 1$, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number $p$. For $h > 1$, we give an XP algorithm with respect to $h+p$. Finally, we consider spine+t-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of $t$ tracks, each of which is a straight line on a separate page, parallel to the spine. In this setting, we can minimize in $2^n \cdot n^{{\mathcal O}(1)}$ time either the number of crossings or, if we disallow crossings, the number of tracks.
Keywords: ordered graphs, book embedding, edge deletion, d-planar, hitting set
Published in DiRROS: 10.12.2024; Views: 56; Downloads: 30
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Abstract: Soil water repellency (SWR) significantly affects plant growth, along with surface and subsurface hydrology, posing a challenge for agricultural productivity and environmental sustainability. Nowadays, the occurrence of microplastics (MP) in the environment, particularly from agricultural practices, raises concerns about MP impact on soil properties. Among them, SWR is affected by hydrophobicity of MP particles detected in soils. This study introduces a method and presents results of a screening test to assess the effects of MP on SWR, utilizing Water Drop Penetration Time (WDPT) analysis under controlled laboratory conditions in destructed soil samples. We compared SWR of two soil types differing in portion of sand, loam and clay. Soils were mixed with three different types of MP originating from agricultural mulch films: low-density polyethylene (LDPE), biodegradable poly butylene adipate terephthalate (PBAT), and starch-based biodegradable plastics (Starch). The MP were milled to a uniform size range of some 10 to 300 μm and mixed with the soil samples. WDPT measurements were taken immediately after mixing and recorded for up to 60 s in order to find MP concentration levels at which strongly or more severely water repellency is inducted on soil samples. Our findings reveal that both, soil type and MP type significantly influence SWR, where there are notable differences observed between bio-based (Starch based) and non-bio-based (LDPE and PBAT) plastics’ effects on SWR in the two tested soil types. Data highlights the distinct behaviour of Starch in altering soil hydrophobicity, prominently different from the impact of both PBAT and LDPE. The measurement technique we have developed for quantifying SWR levels could be used for both research applications and the dissemination of findings. It can significantly enhance decision-making processes regarding the selection of optimal plastic alternatives for agricultural use.
Keywords: environmental pollution, microplastic, water drop penetration time, laboratory test
Published in DiRROS: 10.12.2024; Views: 74; Downloads: 32
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Published in DiRROS: 09.12.2024; Views: 98; Downloads: 36
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Published in DiRROS: 09.12.2024; Views: 78; Downloads: 39
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Keywords: Beremo v gozdnih knjižnicah, promocijski projekti, predstavitev knjige
Published in DiRROS: 06.12.2024; Views: 110; Downloads: 0
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