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

Separation dimension of bounded degree graphs

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.Comment: One result proved in this paper is also present in arXiv:1212.675

Structural parameterizations for boxicity

The boxicity of a graph $G$ is the least integer $d$ such that $G$ has an intersection model of axis-aligned $d$-dimensional boxes. Boxicity, the problem of deciding whether a given graph $G$ has boxicity at most $d$, is NP-complete for every fixed $d \ge 2$. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive $1$-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. that boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 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

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

On the topology Of network fine structures

Multi-relational dynamics are ubiquitous in many complex systems like transportations, social and biological. This thesis studies the two mathematical objects that encapsulate these relationships --- multiplexes and interval graphs. The former is the modern outlook in Network Science to generalize the edges in graphs while the latter was popularized during the 1960s in Graph Theory. Although multiplexes and interval graphs are nearly 50 years apart, their motivations are similar and it is worthwhile to investigate their structural connections and properties. This thesis look into these mathematical objects and presents their connections. For example we will look at the community structures in multiplexes and learn how unstable the detection algorithms are. This can lead researchers to the wrong conclusions. Thus it is important to get formalism precise and this thesis shows that the complexity of interval graphs is an indicator to the precision. However this measure of complexity is a computational hard problem in Graph Theory and in turn we use a heuristic strategy from Network Science to tackle the problem. One of the main contributions of this thesis is the compilation of the disparate literature on these mathematical objects. The novelty of this contribution is in using the statistical tools from population biology to deduce the completeness of this thesis's bibliography. It can also be used as a framework for researchers to quantify the comprehensiveness of their preliminary investigations. From the large body of multiplex research, the thesis focuses on the statistical properties of the projection of multiplexes (the reduction of multi-relational system to a single relationship network). It is important as projection is always used as the baseline for many relevant algorithms and its topology is insightful to understand the dynamics of the system.Open Acces

Matchings, coverings, and Castelnuovo-Mumford regularity

We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio