Chordal graphs with bounded tree-width

TL;DR

It is proved that the number of labelled k -connected chordal graphs with n vertices and tree-width at most t is asymptotically cn − 5 / 2 γ n n !, as n → ∞, for some constants c, γ > 0 depending on t and k.

Abstract

Given $t\geq 2$ and $0\leq k\leq t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is asymptotically $c n^{-5/2} γ^n n!$, as $n\to\infty$, for some constants $c,γ>0$ depending on $t$ and $k$. Additionally, we show that the number of $i$-cliques ($2\leq i\leq t$) in a uniform random $k$-connected chordal graph with tree-width at most $t$ is normally distributed as $n\to\infty$. The asymptotic enumeration of graphs of tree-width at most $t$ is wide open for $t\geq 3$. To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, Graphs and Combinatorics (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on $n$ vertices.

Figures (5)

• Figure 1: Decomposition of a connected graph with tree-width 3 into its 2, 3 and 4-connected components. Vertices with the same label are identified.
• Figure 2: Recursive decomposition of a $k$-connected chordal graph into $(k+1)$-connected components. On the left: a set of graphs from $\mathcal{G}_{k+1}^{(k)}$. On the right: the roots of the graphs are identified and, then, all other $k$-cliques are recursively substituted by graphs in $\mathcal{G}_{k}^{(k)}$ to obtain a grap in $\mathcal{G}_{k}^{(k)}$ whose $(k+1)$-connected components containing the root are the graphs in the set. A label $\mathcal{G}_{k+1}^{(k)}$ or $\mathcal{G}_{k}^{(k)}$ next to a blob means that the blob is any graph from the corresponding class.
• Figure 3: The schema to derive $G_0(\mathbf{x})$ from $G_{t+1}(\mathbf{x})$.
• Figure 4: Tree-decomposition (right) associated to a $2$-connected chordal graph (left) of tree-width 3. The $b$-nodes of the tree represent the $(k+1)$-connected components of the graph, and $c$-nodes represent the $k$-cliques through which the $(k+1)$-connected components are glued together.
• Figure 5: The open region containing 0 delimited by the closed red curve is the $\Delta$-domain $\Delta(\rho(u),\delta,\eta)$, for $\rho(u)\in\mathbb{R}_{>0}$.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1: Dirac D61
• proof
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Lemma 2.5