T-colorings of graphs: recent results and open problems

AbstractSuppose G is a graph and T is a set of nonnegative integers. A T-coloring of G is an assignment of a positive integer ƒ(x) to each vertex x of G so that if x and y are joined by an edge of G, then |ƒ(x) - ƒ(y)ƒ| is not in T. T-colorings were introduced by Hale in connection with the channel assignment problem in communications. Here, the vertices of G are transmitters, an edge represents interference, ƒ(x) is a television or radio channel assigned to x, and T is a set of disallowed separations for channels assigned to interfering transmitters. One seeks to find a T -coloring which minimizes either the number of different channels ƒ(x) used or the distance between the smallest and largest channel. This paper surveys the results and mentions open problems concerned with T-colorings and their variations and generalizations

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

Thinness and its variations on some graph families and coloring graphs of bounded thinness

Interval graphs and proper interval graphs are well known graph classes, for which several generalizations have been proposed in the literature. In this work, we study the (proper) thinness, and several variations, for the classes of cographs, crowns graphs and grid graphs. We provide the exact values for several variants of thinness (proper, independent, complete, precedence, and combinations of them) for the crown graphs $CR_n$. For cographs, we prove that the precedence thinness can be determined in polynomial time. We also improve known bounds for the thinness of $n \times n$ grids $GR_n$ and $m \times n$ grids $GR_{m,n}$, proving that \left \lceil \frac{n-1}{3} \right \rceil \leq \mbox{thin}(GR_n) \leq \left \lceil \frac{n+1}{2} \right \rceil. Regarding the precedence thinness, we prove that \mbox{prec-thin}(GR_{n,2}) = \left \lceil \frac{n+1}{2} \right \rceil and that \left \lceil \frac{n-1}{3} \right \rceil \left \lceil\frac{n-1}{2} \right \rceil + 1 \leq \mbox{prec-thin}(GR_n) \leq \left \lceil\frac{n-1}{2} \right \rceil^2+1. As applications, we show that the $k$-coloring problem is NP-complete for precedence $2$-thin graphs and for proper $2$-thin graphs, when $k$ is part of the input. On the positive side, it is polynomially solvable for precedence proper $2$-thin graphs, given the order and partition

Order of intermittent rock fractured surfaces

According to chaos theory, some underlying patterns can disclose the order of disordered systems. Here, it has been discussed that intermittency of rough rock fractured surfaces is an orderable disorder at intermediate length scales. However, this kind of disorder is more complicated than simple fractal or even multi-scaling behaviours. It is planned to deal with some multifractal spectra that systematically change as a function of the analysed domain. Accordingly, some parameters are introduced that can perfectly take into account such systematic behaviour and quantify the intermittency of the studied surfaces. This framework can be used to quantify and model the roughness of fractured surfaces as a prerequisite factor for the analysis of fluid flow in rock media as well as the shear strength of rock joints. Ultimately, the presented framework can be used for analysing the intermittency of time series and developing new models for predicting, for instance, seismic or flood events in a short time with higher accuracy. © 2022 by the authors

Computable analysis on the space of marked groups

We investigate decision problems for groups described by word problem algorithms. This is equivalent to studying groups described by labelled Cayley graphs. We show that this corresponds to the study of computable analysis on the space of marked groups, and point out several results of computable analysis that can be directly applied to obtain group theoretical results. Those results, used in conjunction with the version of Higman's Embedding Theorem that preserves solvability of the word problem, provide powerful tools to build finitely presented groups with solvable word problem but with various undecidable properties. We also investigate the first levels of an effective Borel hierarchy on the space of marked groups, and show that on many group properties usually considered, this effective hierarchy corresponds sharply to the Borel hierarchy. Finally, we prove that the space of marked groups is a Polish space that is not effectively Polish. Because of this, many of the most important results of computable analysis cannot be applied to the space of marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and a theorem of Moschovakis. The space of marked groups constitutes the first natural example of a Polish space that is not effectively Polish.Comment: 46 pages, Theorem 4.6 was false as stated, it appears now, having been corrected, as Theorem 5.

Adventures in Semantic Publishing: Exemplar Semantic Enhancements of a Research Article

Scientific innovation depends on finding, integrating, and re-using the products of previous research. Here we explore how recent developments in Web technology, particularly those related to the publication of data and metadata, might assist that process by providing semantic enhancements to journal articles within the mainstream process of scholarly journal publishing. We exemplify this by describing semantic enhancements we have made to a recent biomedical research article taken from PLoS Neglected Tropical Diseases, providing enrichment to its content and increased access to datasets within it. These semantic enhancements include provision of live DOIs and hyperlinks; semantic markup of textual terms, with links to relevant third-party information resources; interactive figures; a re-orderable reference list; a document summary containing a study summary, a tag cloud, and a citation analysis; and two novel types of semantic enrichment: the first, a Supporting Claims Tooltip to permit “Citations in Context”, and the second, Tag Trees that bring together semantically related terms. In addition, we have published downloadable spreadsheets containing data from within tables and figures, have enriched these with provenance information, and have demonstrated various types of data fusion (mashups) with results from other research articles and with Google Maps. We have also published machine-readable RDF metadata both about the article and about the references it cites, for which we developed a Citation Typing Ontology, CiTO (http://purl.org/net/cito/). The enhanced article, which is available at http://dx.doi.org/10.1371/journal.pntd.0000228.x001, presents a compelling existence proof of the possibilities of semantic publication. We hope the showcase of examples and ideas it contains, described in this paper, will excite the imaginations of researchers and publishers, stimulating them to explore the possibilities of semantic publishing for their own research articles, and thereby break down present barriers to the discovery and re-use of information within traditional modes of scholarly communication