Size-Adaptive Texture Atlas Generation and Remapping for 3D Urban Building Models

A high-fidelity 3D urban building model requires large quantities of detailed textures, which can be non-tiled or tiled ones. The fast loading and rendering of these models remain challenges in web-based large-scale 3D city visualization. The traditional texture atlas methods compress all the textures of a model into one atlas, which needs extra blank space, and the size of the atlas is uncontrollable. This paper introduces a size-adaptive texture atlas method that can pack all the textures of a model without losing accuracy and increasing extra storage space. Our method includes two major steps: texture atlas generation and texture atlas remapping. First, all the textures of a model are classified into non-tiled and tiled ones. The maximum supported size of the texture is acquired from the graphics hardware card, and all the textures are packed into one or more atlases. Then, the texture atlases are remapped onto the geometric meshes. For the triangle with the original non-tiled texture, new texture coordinates in the texture atlases can be calculated directly. However, as for the triangle with the original tiled texture, it is clipped into many unit triangles to apply texture mapping. Although the method increases the mesh vertex number, the increased geometric vertices have much less impact on the rendering efficiency compared with the method of increasing the texture space. The experiment results show that our method can significantly improve building model rendering efficiency for large-scale 3D city visualization.

Note on Reliability Analysis of Cartesian Product of Networks for Components

Connectivity is a critical parameter which can measure the reliability of networks. Let [Formula: see text] be a vertex set of [Formula: see text]. If [Formula: see text] has at least [Formula: see text] components, then [Formula: see text] is a [Formula: see text]-component cut of [Formula: see text]. The [Formula: see text]-component connectivity [Formula: see text] of [Formula: see text] is the vertex number of a smallest [Formula: see text]-component cut. Cartesian product of graphs is a useful method to construct a large network. We will use Cauchy–Schwarz inequality to determine the component connectivity of Cartesian product of some graphs.

Finding Temporal Paths Under Waiting Time Constraints

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $$\varDelta $$ Δ , referred to as $$\varDelta $$ Δ -restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the “restless variant” of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

Feedback Arc Number and Feedback Vertex Number of Cartesian Product of Directed Cycles

For a digraph D, the feedback vertex number τD, (resp. the feedback arc number τ′D) is the minimum number of vertices, (resp. arcs) whose removal leaves the resultant digraph free of directed cycles. In this note, we determine τD and τ′D for the Cartesian product of directed cycles D=Cn1→□Cn2→□…Cnk→. Actually, it is shown that τ′D=n1n2…nk∑i=1k1/ni, and if nk≥…≥n1≥3 then τD=n2…nk.

Which properties of the visual stimuli predict the type of representation used in mental rotation?

Two modes of internal representation, holistic and piecemeal transformation, have been reported as a means to perform mental rotation (MR) tasks. The stimulus complexity effect has been proposed as an indicator to disentangle between these two representation types. However, the complexity effect has not been fully confirmed owing to the fact that different performances could result from different types of stimuli. Moreover, whether the non-mirror foils play a role in forcing participants to encode all the information from the stimuli in MR tasks is still under debate. This study aims at testing the association between these two common types of representation with different stimuli in MR tasks. First, the numbers of segments and vertices in polygon stimuli were manipulated to test which property of the visual stimuli is more likely to influence the representation in MR tasks. Second, the role of non-mirror foils was examined by comparing the stimulus complexity effect in both with- and without-non-mirror foils conditions. The results revealed that the segment number affected the slope of the linear function relating response times to rotation angle, but the vertex number in the polygons did not. This suggests that a holistic representation was more likely to be adopted in processing integrated objects, whereas a piecemeal transformation was at play in processing multi-part objects. In addition, the stimulus complexity effect was observed in the with-non-mirror foils condition but not in the without-non-mirror foils one, providing a direct evidence to support the role of non-mirror foils in MR tasks.

Matching Number, Independence Number, and Covering Vertex Number of Γ(Zn)

Graph invariants are the properties of graphs that do not change under graph isomorphisms, the independent set decision problem, vertex covering problem, and matching number problem are known to be NP-Hard, and hence it is not believed that there are efficient algorithms for solving them. In this paper, the graph invariants matching number, vertex covering number, and independence number for the zero-divisor graph over the rings Z p k and Z p k q r are determined in terms of the sets S p i and S p i q j respectively. Accordingly, a formula in terms of p , q , k , and r, with n = p k , n = p k q r is provided.

Correction: A tetramer micelle: the smallest aggregation number corresponding to the vertex number of regular polyhedra in platonic micelles

Correction for ‘A tetramer micelle: the smallest aggregation number corresponding to the vertex number of regular polyhedra in platonic micelles’ by Shota Fujii et al., Soft Matter, 2018, DOI: 10.1039/c7sm02028g.

A tetramer micelle: the smallest aggregation number corresponding to the vertex number of regular polyhedra in platonic micelles

Sulfonatocalix[4]arene-based amphiphiles form monodisperse tetramer micelles whose aggregation number is the smallest corresponding to the vertex number of regular polyhedra.

Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph

Let G? be an oriented graph and S(G?) be its skew-adjacency matrix, where G is called the underlying graph of G?. The skew-rank of G?, denoted by sr(G?), is the rank of S(G?). Denote by d(G) = |E(G)|-|V(G)| + ?(G) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and ?(G) are the edge number, vertex number and the number of connected components of G, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76-86] proved that sr(G?) ? r(G) + 2d(G) for an oriented graph G?, where r(G) is the rank of the adjacency matrix of G, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of sr(G?) of an oriented graph G? in terms of r(G) and d(G) of its underlying graph G is left open till now. In this paper, we prove that sr(G?) ? r(G)-2d(G) for an oriented graph G? and characterize the graphs whose skew-rank attain the lower bound.