Nordhaus-Gaddum for Treewidth

We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.

💡 Research Summary

The paper establishes a Nordhaus‑Gaddum‑type inequality for the graph parameter treewidth. The main result (Theorem 1) states that for every n‑vertex graph G, the sum of the treewidth of G and the treewidth of its complement (\overline{G}) is at least (n-2). This lower bound is shown to be tight.

The proof hinges on a key auxiliary lemma (Lemma 2): if a graph on n vertices contains no induced 4‑cycle and has no clique of size larger than k, then its treewidth is at least (n-k). The authors prove this by constructing a bramble from the set of edges, showing that any hitting set must have size at least (n-k+1), and then invoking the Treewidth Duality Theorem (Seymour‑Thomas) which equates treewidth with the order of a maximal bramble.

To prove Theorem 1, let (k = \operatorname{tw}(G)). The authors embed G in a spanning k‑tree H. Because H is chordal, it contains no induced 4‑cycle, and by construction it lacks a ((k+2))-clique. Applying Lemma 2 to H yields (\operatorname{tw}(H) \ge n-k-2). Since H spans G, (\operatorname{tw}(G) \ge n-k-2). Repeating the argument for the complement gives (\operatorname{tw}(\overline{G}) \ge k). Adding the two inequalities gives the desired bound ( \operatorname{tw}(G)+\operatorname{tw}(\overline{G}) \ge n-2).

The authors then demonstrate tightness. They define a special k‑tree (Q_{k,n}) consisting of a k‑clique together with (n-k) vertices each adjacent to all vertices of the clique. For this family, (\operatorname{tw}(Q_{k,n}) = n-k-1) and (\operatorname{tw}(\overline{Q_{k,n}}) = k), so the sum equals (n-1), exceeding the general lower bound. For all other k‑trees, the bound (n-2) is attained, establishing that the inequality cannot be improved in general.

Further consequences are explored. Theorem 3 shows that any graph of girth at least five satisfies (\operatorname{tw}(G) \ge n-3). The paper also discusses random graphs: using results of Perarnau and Serra, it is noted that for (G\in G(n,1/2)) the treewidth is almost surely (n-o(n)), implying that the trivial upper bound (\operatorname{tw}(G)+\operatorname{tw}(\overline{G})\le 2n-2) is asymptotically tight (the sum is (2n-o(n)) with high probability).

Proposition 5 provides a more general statement about the union of two graphs: the union cannot contain a clique larger than (\operatorname{tw}(G_1)+\operatorname{tw}(G_2)+2) vertices, and this bound is sharp. This leads to questions about the chromatic number of such unions, drawing parallels with Ringel’s earth‑moon problem concerning the chromatic number of the union of two planar graphs. A simple greedy algorithm yields an upper bound (\chi(G_1\cup G_2)\le 4k) when (\operatorname{tw}(G_1),\operatorname{tw}(G_2)\le k).

Overall, the paper contributes a clean, optimal Nordhaus‑Gaddum inequality for treewidth, connects it to bramble theory and chordal graph structure, and opens several avenues for further research on graph unions, coloring, and random graph behavior.