arXiv 0803.36702008-03-26cs.DM

关于AT-free图和圆弧图的立方性

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

L. Sunil Chandran, Mathew C. Francis, Naveen Sivadasan

备注：: 9 pages, 0 figures

摘要：

$k$维中的单位立方体（$k$ -立方体）被定义为笛卡尔积$R_1\times R_2\times...\times R_k$，其中$R_i$（对于$1\leq i\leq k$）是实数线上形式为$[a_i,a_i+1]$的闭区间。如果$G$的每个顶点都可以映射到一个$k$ -立方体上，使得当且仅当它们对应的$k$-立方体有一个非空的交集时，两个顶点在$G$中相邻，则称$G$在$n$节点上的图$G$可表示为$k$-立方体的交集（$k$维中的立方体表示）。表示为$\cubi(G)$的$G$的\emph{cubicity}是最小$k$，对于该最小$k$，$G$可以表示为$k$ -立方体的交集。我们给出了一个$O(bw\cdot n)$算法来计算一般图$G$在$bw+1$维上的立方体表示，其中$bw$是$G$的\emph{bandwidth}.作为结果，我们得到了许多著名的图类，如AT-自由图，圆弧图和共同可比图的$O(Δ)$带宽的立方度的$O(Δ)$上界。因此我们有：1）$\cubi(G)\leq 3Δ-1$，如果$G$是一个AT-自由图。2)$\cubi(G)\leq 2Δ+1$，如果$G$是圆弧图。3)$\cubi(G)\leq 2Δ$，如果$G$是协同可比性图。同样对于这些图形类，存在用于带宽计算的常数因子近似算法，其生成具有$O(Δ)$宽度的顶点的排序。因此，我们可以在多项式时间内生成$O(Δ)$维的这种图的立方体表示。

英文摘要：

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 $O(Δ)$ 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 $O(Δ)$ bandwidth. Thus we have: 1) $\cubi(G)\leq 3Δ-1$, if $G$ is an AT-free graph. 2) $\cubi(G)\leq 2Δ+1$, if $G$ is a circular-arc graph. 3) $\cubi(G)\leq 2Δ$, 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 $O(Δ)$ width. We can thus generate the cube representation of such graphs in $O(Δ)$ dimensions in polynomial time.

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