It is known that any chordal graph on n vertices can be represented as the intersection of n subtrees in a tree on n nodes (Gavril in J Comb Theory 16:47–56, 1974). This characterization has been recently used to generate random chordal graphs on n vertices by generating n subtrees of a tree on n nodes. The space (and thus time) complexity of an algorithm generating n subtrees of a tree on n nodes is at least the sum of the sizes of the generated subtrees. The determination of this sum was left as an open question in Şeker et al. (Generation of random chordal graphs using subtrees of a tree. arXiv preprint arXiv:1810.13326, 2018). In this paper, we show that the sum of the sizes of n subtrees in a tree on n nodes is \(\varTheta (m\sqrt{n})\). We also show that we can confine ourselves to contraction-minimal subtree intersection representations because they are sufficient to generate every chordal graph with strictly positive probability. Moreover, the sum of the sizes of the subtrees in a contraction-minimal representation is at most \(2m+n\). We use this result to derive the first linear-time random chordal graph generator. Based on contraction-minimal representations, we also derive connectivity-related structural properties of chordal graphs. Besides these theoretical results, we also conduct experiments to study the quality of the chordal graphs generated by our algorithm and compare them to those generated by existing methods from the literature. Our algorithm does not generate chordal graphs uniformly at random, which is a quite challenging open question, irrespective of the time complexity of the generator. However, our experimental study suggests that the generated graphs have a fairly varied structure as indicated by the sizes of maximal cliques. Furthermore, our algorithm is simple to implement and produces graphs with 10,000 vertices and \(4 \times 10^7\) edges in less than one second on a laptop computer.

众所周知，在n个顶点上的任何弦图都可以表示为n个节点上的树中n个子树的交集（1974年J Comb Theory 16：47-56中的Gavril）。通过在n个节点上生成树的n个子树，最近已将此特征用于在n个顶点上生成随机弦弦图。在n上生成树的n个子树的算法的空间（以及时间）复杂度节点至少是所生成子树大小的总和。在Şeker等人中，这个总和的确定是一个悬而未决的问题。（使用树的子树生成随机和弦图.arXiv预印本arXiv：1810.13326,2018年）。在本文中，我们表明，的尺寸总和Ñ子树的树上Ñ节点是\（\ varTheta（米\ SQRT {N}）\） 。我们还表明，我们可以将自己局限于收缩-最小子树相交表示，因为它们足以生成严格正概率的每个弦图。此外，最小收缩表示形式的子树大小之和最多为\（2m + n \）。我们用这个结果来推导第一个线性时间随机弦图发生器。基于收缩最小表示，我们还导出了弦图的与连通性相关的结构特性。除了这些理论结果外，我们还进行实验以研究由我们的算法生成的弦图的质量，并将其与文献中现有方法生成的弦图进行比较。我们的算法不会随机均匀地生成和弦图，这是一个非常具有挑战性的开放性问题，而与生成器的时间复杂度无关。但是，我们的实验研究表明，生成的图具有最大的团簇大小所指示的相当变化的结构。此外，我们的算法易于实现，可以生成具有10,000个顶点的图形，并且\（4 \ times 10 ^ 7 \）在便携式计算机上的边缘不到一秒钟。