On the cubicity of AT-free graphs and circular-arc graphs

TL;DR

This paper introduces an algorithm to compute the cubicity of graphs based on bandwidth, providing tight bounds for classes like AT-free, circular-arc, and co-comparability graphs, and enabling efficient cube representations.

Contribution

The paper presents an $O(bw imes n)$ algorithm for cube representation using bandwidth, establishing new upper bounds on cubicity for specific graph classes and offering polynomial-time approximation algorithms.

Findings

Cubicity of AT-free graphs is at most $3\Delta - 1$.

Circular-arc graphs have cubicity at most $2\Delta + 1$.

Co-comparability graphs have cubicity at most $2\Delta$.

Abstract

A unit cube in $k$ dimensions ($k$-cube) is defined as the the Cartesian product $R_1\times R_2\times...\times R_k$ where $R_i$(for $1\leq i\leq k$) is a closed interval of the form $[a_i,a_i+1]$ on the real line. A graph $G$ on $n$ nodes is said to be representable as the intersection of $k$-cubes (cube representation in $k$ dimensions) if each vertex of $G$ can be mapped to a $k$-cube such that two vertices are adjacent in $G$ if and only if their corresponding $k$-cubes have a non-empty intersection. The \emph{cubicity} of $G$ denoted as $\cubi(G)$ is the minimum $k$ for which $G$ can be represented as the intersection of $k$-cubes. We give an $O(bw\cdot n)$ algorithm to compute the cube representation of a general graph $G$ in $bw+1$ dimensions given a bandwidth ordering of the vertices of $G$, where $bw$ is the \emph{bandwidth} of $G$. As a consequence, we get $O(\Delta)$ upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and co-comparability graphs which have $O(\Delta)$ bandwidth. Thus we have: 1) $\cubi(G)\leq 3\Delta-1$, if $G$ is an AT-free graph. 2) $\cubi(G)\leq 2\Delta+1$, if $G$ is a circular-arc graph. 3) $\cubi(G)\leq 2\Delta$, if $G$ is a co-comparability graph. Also for these graph classes, there are constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with $O(\Delta)$ width. We can thus generate the cube representation of such graphs in $O(\Delta)$ dimensions in polynomial time.