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

Dimension of CPT posets

A collection of linear orders on $X$, say $\mathcal{L}$, is said to \emph{realize} a partially ordered set (or poset) $\mathcal{P} = (X, \preceq)$ if, for any two distinct $x,y \in X$, $x \preceq y$ if and only if $x \prec_L y$, $\forall L \in \mathcal{L}$. We call $\mathcal{L}$ a \emph{realizer} of $\mathcal{P}$. The \emph{dimension} of $\mathcal{P}$, denoted by $dim(\mathcal{P})$, is the minimum cardinality of a realizer of $\mathcal{P}$. A \emph{containment model} $M_{\mathcal{P}}$ of a poset $\mathcal{P}=(X,\preceq)$ maps every $x \in X$ to a set $M_x$ such that, for every distinct $x,y \in X,\ x \preceq y$ if and only if $M_x \varsubsetneq M_y$. We shall be using the collection $(M_x)_{x \in X}$ to identify the containment model $M_{\mathcal{P}}$. A poset $\mathcal{P}=(X,\preceq)$ is a Containment order of Paths in a Tree (CPT poset), if it admits a containment model $M_{\mathcal{P}}=(P_x)_{x \in X}$ where every $P_x$ is a path of a tree $T$, which is called the host tree of the model. We show that if a poset $\mathcal{P}$ admits a CPT model in a host tree $T$ of maximum degree $\Delta$ and radius $r$, then \rogers{$dim(\mathcal{P}) \leq \lg\lg \Delta + (\frac{1}{2} + o(1))\lg\lg\lg \Delta + \lg r + \frac{1}{2} \lg\lg r + \frac{1}{2}\lg \pi + 3$. This bound is asymptotically tight up to an additive factor of $\min(\frac{1}{2}\lg\lg\lg \Delta, \frac{1}{2}\lg\lg r)$. Further, let $\mathcal{P}(1,2;n)$ be the poset consisting of all the $1$-element and $2$-element subsets of $[n]$ under `containment' relation and let $dim(1,2;n)$ denote its dimension. The proof of our main theorem gives a simple algorithm to construct a realizer for $\mathcal{P}(1,2;n)$ whose cardinality is only an additive factor of at most $\frac{3}{2}$ away from the optimum.Comment: 10 Page

Local boxicity and maximum degree

The local boxicity of a graph G, denoted by lbox(G), is the minimum positive integer l such that G can be obtained using the intersection of k (where k≥l) interval graphs where each vertex of G appears as a non-universal vertex in at most l of these interval graphs. Let G be a graph on n vertices having m edges. Let Δ denote the maximum degree of a vertex in G. We show that, • lbox(G)≤213logΔ. • lbox(G)∈O([Formula presented]). • lbox(G)≤(213log+2)m. • the local boxicity of G is at most its product dimension. This connection helps us in showing that the local boxicity of the Kneser graph K(n,k) is at most [Formula presented]log⁡log⁡n. The above results can be extended to the local dimension of a partially ordered set due to the known connection between local boxicity and local dimension. Using this connection along with known results it can be shown that there exist graphs of maximum degree Δ having a local boxicity of Ω([Formula presented]). There also exist graphs on n vertices and graphs on m edges having local boxicity of Ω([Formula presented]) and Ω([Formula presented]), respectively. © 2022 Elsevier B.V

Bounding Threshold Dimension: Realizing Graphic Boolean Functions as the AND of Majority Gates

A graph G on n vertices is a threshold graph if there exist real numbers $$a:1,a_2, \ldots, a_n$$ and b such that the zero-one solutions of the linear inequality $$\sum \limits _{i=1}^n a_i x_i \le b$$ are the characteristic vectors of the cliques of G. Introduced in [Aggregation of inequalities in integer programming. Chvátal and Hammer, Annals of Discrete Mathematics, 1977], the threshold dimension of a graph G, denoted by $$\textrm{dim}:{\textrm{TH}}(G)$$, is the minimum number of threshold graphs whose intersection yields G. Given a graph G on n vertices, in line with Chvátal and Hammer, $$f:G_\{0,1\}^n \rightarrow \{0,1\}$$ is the Boolean function that has the property that $$f:G(x) = 1$$ if and only if x is the characteristic vector of a clique in G. A Boolean function f for which there exists a graph G such that $$f=f:G$$ is called a graphic Boolean function. It follows that for a graph G, $$\textrm{dim}:{\textrm{TH}}(G)$$ is precisely the minimum number of majority gates whose AND (or conjunction) realizes the graphic Boolean function $$f:G$$. The fact that there exist Boolean functions which can be realized as the AND of only exponentially many majority gates motivates us to study threshold dimension of graphs. We give tight or nearly tight upper bounds for the threshold dimension of a graph in terms of its treewidth, maximum degree, degeneracy, number of vertices, size of a minimum vertex cover, etc. We also study threshold dimension of random graphs and graphs with high girth. © 2022, Springer Nature Switzerland AG

Bounding threshold dimension: realizing graphic Boolean functions as the AND of majority gates

A graph $G$ on $n$ vertices is a \emph{threshold graph} if there exist real numbers $a_1,a_2, \ldots, a_n$ and $b$ such that the zero-one solutions of the linear inequality $\sum \limits_{i=1}^n a_i x_i \leq b$ are the characteristic vectors of the cliques of $G$. Introduced in [Chv{\'a}tal and Hammer, Annals of Discrete Mathematics, 1977], the \emph{threshold dimension} of a graph $G$, denoted by \dimth(G), is the minimum number of threshold graphs whose intersection yields $G$. Given a graph $G$ on $n$ vertices, in line with Chv{\'a}tal and Hammer, $f_G\colon \{0,1\}^n \rightarrow \{0,1\}$ is the Boolean function that has the property that $f_G(x) = 1$ if and only if $x$ is the characteristic vector of a clique in $G$. A Boolean function $f$ for which there exists a graph $G$ such that $f=f_G$ is called a \emph{graphic} Boolean function. It follows that for a graph $G$, \dimth(G) is precisely the minimum number of \emph{majority} gates whose AND (or conjunction) realizes the graphic Boolean function $f_G$. The fact that there exist Boolean functions which can be realized as the AND of only exponentially many majority gates motivates us to study threshold dimension of graphs. We give tight or nearly tight upper bounds for the threshold dimension of a graph in terms of its treewidth, maximum degree, degeneracy, number of vertices, size of a minimum vertex cover, etc. We also study threshold dimension of random graphs and graphs with high girth