Treewidth Analysis of Outerstring Graphs

• Treewidth of outerstring graphs is defined via tree decompositions and is bounded linearly by the Hadwiger number, as shown by the inequality tw(G) ≤ 15 ⋅ had(G) − 2.
• The analysis also highlights logarithmic treewidth bounds in sparse cases using arboricity and biclique exclusion, yielding efficient algorithms for NP-hard problems.
• Advanced techniques involving geometric separators and degeneracy measures underpin robust algorithmic strategies and structural insights for outerstring graphs.

An outerstring graph is the intersection graph of strings (simple arcs or curves), each drawn inside a closed disk with precisely one endpoint on the disk boundary. Treewidth for such graphs has emerged as a central parameter, governing both structural and algorithmic properties. Key results provide precise and explicit upper and lower bounds for the treewidth of outerstring graphs in terms of combinatorial invariants such as the Hadwiger number, arboricity, and degeneracy. Deep connections with separator theorems, minor theory, and algorithmic tractability have been established, and outerstring graphs serve as canonical objects in graph minor theory and geometric intersection graph research.

1. Definitions and Basic Properties

An outerstring graph $G$ is the intersection graph of a collection $\mathcal{S} = \{S_1, \ldots, S_n\}$ of simple arcs in a closed disk %%%%2%%%% such that each $S_i$ has exactly one endpoint (the root) on $\partial\Delta$, with distinct roots for distinct strings. Vertices $i,j$ are adjacent if and only if the interiors of $S_i$ and $S_j$ intersect.

Treewidth $\operatorname{tw}(G)$ is the minimum width among tree decompositions of $G$: if $(T,\chi)$ is a tree decomposition, width is $\max_{t\in V(T)} |\chi(t)|-1$.

The Hadwiger number $h=\operatorname{had}(G)$ is the largest $t$ such that $K_t \preceq G$ (i.e., $K_t$ is a minor of $G$).

Other invariants of relevance are:

• Arboricity $\alpha(G)$: the minimum number of forests which partition the edge set.
• (t,d)-degeneracy: $G$ has an ordered $t$-colouring so that each vertex has at most $d$ higher-colour neighbours.

These notions extend naturally to outerstring graphs in higher-genus surfaces and to those with multiple grounding disks (the $(g,c)$-outerstring generalization) (Campbell et al., 25 Oct 2024, Karol, 19 Nov 2025).

2. Linear Treewidth–Hadwiger Number Bound

A cornerstone result is the explicit linear relationship between the treewidth and the Hadwiger number for all outerstring graphs:

$$\operatorname{tw}(G) \leq 15\cdot \operatorname{had}(G) - 2.$$

This inequality is uniform over all outerstring graphs. The proof proceeds by constructing small balanced separators among the root points on $\partial\Delta$, utilizing Menger’s theorem and geometric constraints to prohibit large clique minors unless treewidth is large. Specifically, if $G$ excludes $K_{t+1}$ as a minor, every sufficiently large subset of roots on $\partial\Delta$ can be separated with $O(t)$ vertices. Transforming balanced separator existence into a treewidth bound via standard separator-to-treewidth lemmas yields the claimed linear form (Campbell et al., 25 Oct 2024).

This result supports the broader conjecture that for all (tw, had)-bounded classes—i.e., those excluding some planar graph as induced minor—treewidth is linearly bounded by the Hadwiger number.

3. Arboricity, Biclique Exclusion, and Logarithmic Bounds

Beyond the treewidth–Hadwiger linear upper bound, a different regime emerges for sparse outerstring graphs. For $n$-vertex outerstring graphs with arboricity $\alpha$, the treewidth satisfies:

$\operatorname{tw}(G) = O(\alpha\log n)$

with a lower bound of $\Omega(\alpha\log(n/\alpha))$, both realized by explicit constructions (An et al., 25 Jun 2024). This bound is sharp up to constants even for outersegment graphs.

For $t$-biclique-free graphs (i.e., excluding $K_{t,t}$ as a subgraph), a folklore fact is that arboricity is $O(t\log t)$, leading to: $\operatorname{tw}(G) = O(t\log t \log n)$ for such outerstring graphs. These results extend to wider classes of intersection graphs, and are established by combining geometric separator arguments (crossing levels, layer reductions) with bramble–well-linked set techniques for finding large clique minors.

4. Degeneracy, Degree Bounds, and Extensions to Surfaces

Recent work leverages ordered $(t,d)$-degeneracy as a finer measure of complexity. An outerstring graph $G$ in the plane with maximum degree $\Delta$ is $(\Delta+1,\Delta)$-degenerate, leading to the explicit quadratic bound:

$\operatorname{tw}(G) \leq 3\Delta^2 + 5\Delta + 1.$

The methodology constructs a "coloured planarisation" of the curve arrangement, with crossings replaced by dummy vertices and monochromatic runs contracted. The strong product structure

$G \lesssim H \boxtimes K_{d+1},$

where $H$ is a planar (or bounded-genus) graph of bounded treewidth, allows the application of classical results (Robertson–Seymour III) on the treewidth of graphs of bounded radius and genus (Karol, 19 Nov 2025). In the $(g,c)$-outerstring setting, the bound becomes:

$$\operatorname{tw}(G) \leq (2t-1) c (2g+3) (d+1) - 1.$$

This generalization covers intersection graphs grounded on multiple disks in surfaces of Euler genus $g$.

5. Algorithmic Consequences and Applications

The explicit treewidth bounds for outerstring graphs underpin a wealth of algorithmic results:

• Polynomial-time algorithms: For any fixed biclique size $t$, all standard NP-complete problems, such as Independent Set, Vertex Cover, Dominating Set, Feedback Vertex Set, and $q$-Coloring, are solvable in polynomial time for $K_{t,t}$-free outerstring graphs since $\operatorname{tw}(G) = O(\log n)$. The algorithms are "robust" and do not require the geometric representation (An et al., 25 Jun 2024).
• Subexponential algorithms: For arbitrary (possibly dense) outerstring graphs, subexponential time algorithms for problems like Vertex Cover and Feedback Vertex Set are achievable via branching on bicliques in time $2^{O(\sqrt{k} \log^2 k)} n^{O(1)}$, where $k$ is the solution size.
• General MSO-definable problems: For bounded-degeneracy (or bounded degree) outerstring graphs, classical meta-theorems (Courcelle's theorem) yield $f(k) n^{O(1)}$-time algorithms for all monadic second-order properties, leveraging the existence of tree decompositions of the proven width (Karol, 19 Nov 2025).
• Separator theorems and layout parameters: Balanced separators of size $O(\Delta)$ for degree-$\Delta$ graphs imply small bandwidth, queue number, and other layout invariants, as well as fast approximation for packing and coloring problems.
• Certifying decompositions: Since recognition of outerstring graphs is NP-hard, it is of interest to find certifiable treewidth decompositions via combinatorial means.

6. Limitations and Extremal Constructions

While outerstring graphs exhibit strong treewidth–minor structure, the relationship fails for more general classes. In particular, explicit layered-wheel-like graphs of arbitrarily large treewidth exist such that every outerstring induced subgraph is of bounded treewidth (absolute constant $L$), contradicting a former conjecture of Trotignon that such obstructions do not arise when induced minors of large grids and bicliques are excluded (Chudnovsky et al., 8 Jul 2025). These constructions show:

• For every $k,g$ there exists a graph $G^g_k$ of treewidth $\geq k-1$ with every outerstring induced subgraph having $\operatorname{tw}\leq L$ (some large constant, explicit bound given).
• The mechanism ensures that no long induced thetas occur (a forbidden subgraph for outerstring graphs), while still spacing degree-4 vertices and allowing arbitrarily large girth.

This shows that bounding the treewidth of outerstring induced subgraphs alone does not suffice to guarantee overall treewidth bounds in classes without large grid or biclique induced minors.

7. Open Problems and Further Directions

Several questions remain open:

• Is the leading constant in the outerstring linear treewidth–Hadwiger bound optimal, or can it be substantially reduced?
• Does every hereditary class excluding a planar induced minor admit linear treewidth–Hadwiger bounds?
• Can the hidden $(\log t)$ terms in the biclique-free logarithmic bound be removed or reduced?
• Which other geometric graph classes admit treewidth bounds of the form $\operatorname{tw}(G)=O(\operatorname{had}(G))$ or $\operatorname{tw}(G)=O(\alpha\log n)$?
• Is it possible to systematically certify treewidth-small decompositions for outerstring graphs in a fully combinatorial manner, circumventing the need for a geometric representation?

Outerstring graphs represent one of the most tractable intersection graph families for treewidth analysis, supporting robust algorithmic applications and serving as a template for extending structural graph theory into geometric and topological settings (Campbell et al., 25 Oct 2024, An et al., 25 Jun 2024, Karol, 19 Nov 2025, Chudnovsky et al., 8 Jul 2025).