On local structures of cubicity 2 graphs

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the $X$-axis, distance $1 + \epsilon$ ($0 < \epsilon < 1$) apart. This family of graphs allow us to study local structures of unit square intersection graphs, that is, graphs with cubicity 2. The complexity of determining whether a tree has cubicity 2 is unknown while the graph recognition problem for unit square intersection graph is known to be NP-hard. We present a polynomial time algorithm for recognizing trees that admit a 2SUIG representation

Local Boxicity, Local Dimension, and Maximum Degree

In this paper, we focus on two recently introduced parameters in the literature, namely `local boxicity' (a parameter on graphs) and `local dimension' (a parameter on partially ordered sets). We give an `almost linear' upper bound for both the parameters in terms of the maximum degree of a graph (for local dimension we consider the comparability graph of a poset). Further, we give an $O(n\Delta^2)$ time deterministic algorithm to compute a local box representation of dimension at most $3\Delta$ for a claw-free graph, where $n$ and $\Delta$ denote the number of vertices and the maximum degree, respectively, of the graph under consideration. We also prove two other upper bounds for the local boxicity of a graph, one in terms of the number of vertices and the other in terms of the number of edges. Finally, we show that the local boxicity of a graph is upper bounded by its `product dimension'.Comment: 11 page

Revisiting Interval Graphs for Network Science

The vertices of an interval graph represent intervals over a real line where overlapping intervals denote that their corresponding vertices are adjacent. This implies that the vertices are measurable by a metric and there exists a linear structure in the system. The generalization is an embedding of a graph onto a multi-dimensional Euclidean space and it was used by scientists to study the multi-relational complexity of ecology. However the research went out of fashion in the 1980s and was not revisited when Network Science recently expressed interests with multi-relational networks known as multiplexes. This paper studies interval graphs from the perspective of Network Science

New insights into water's phase diagram using ammonium fluoride

Ice is a complex, yet highly relevant material and has been a rife area of research since the beginning of the 20th century.1-5 Understanding ice is expected to have consequences not just for furthering our appreciation of the different states of water, but also general chemistry, physics and geology.2, 6, 7 It has often been found that properties first observed in ice (e.g. stacking disorder) are also present in other materials.2, 6, 8, 9 This thesis largely builds on work performed by Shephard et al. which explored the effect of 2.5 mol% NH4F in ice, and astoundingly fully prevented ice II formation.10 Initially, the thesis focuses on the effect of adding NH4F to ice at ambient pressure, which is demonstrated to produce a denser material than pure ice. At 0.5 GPa, NH4F-ice solid solutions (≥ 12 mol%) surprisingly produce stable ice XII-type structures. Additionally, upon the mapping of the 2.5 mol% NH4F phase diagram to 1.7 GPa, it was found that phase-pure ice XII could be quenched at 1.1 GPa. Both ice XIItype structures did not require an amorphous precursor. The influence of NH4F in ice is explored in mixtures that are subjected to the compression conditions that yield high-density amorphous ice ‘pressure-induced amorphised’ upon their compression to 1.4 GPa at 77 K. Unexpectedly, the crossover of PIA to recrystallisation is determined as beginning on the water-rich side (35 mol% NH4F) of the solid solutions. Stacking disorder from the heating of NH4F II and III at ambient pressure is quantified. The materials reach a maximum cubicity of 77%, yet the stacking disorder obtained from each material is unique. Remarkably NH4F III did not transform to an amorphous phase upon heating. The final standalone chapter focuses on the ordering of ices V/XIII and IX with 0.01 M HCl doping

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement