String graphs are quasi-isometric to planar graphs

TL;DR

This paper resolves whether string graphs share the large-scale geometry of planar graphs by proving a strong quasi-isometry: for every countable string graph $S$ there exists a planar graph $G$ with $V(G)=V(S)$ such that $\frac{1}{23660800} d_S(u,v) \le d_G(u,v) \le 162 d_S(u,v)$ for all $u,v$. The authors develop an extensive coarse-geometry framework—region intersection graphs, impressions, encasings, cages, and fortifications—to transfer distance control between string-graph representations and planar minors, with a finite-case proof that extends to countable graphs. The main contributions include the finite-string-graph theorem with explicit Lipschitz constants, its extension to all string graphs, and broad corollaries: string graphs have Assouad–Nagata dimension at most $2$, are accessible in the quasi-transitive setting, and many group-theoretic characterizations follow from the quasi-isometry to planar graphs. The results thereby extend planar-graph structure and algorithmic techniques to the broader class of string graphs, and yield polynomial-time constructions for finite representations, as well as implications for planar metric graphs and complete Riemannian planes.

Abstract

We prove that for every countable string graph $S$, there is a planar graph $G$ with $V(G)=V(S)$ such that \[ \frac{1}{23660800}d_S(u,v) \le d_G(u,v) \le 162 d_S(u,v) \] for all $u,v\in V(S)$, where $d_S(u,v)$, $d_G(u,v)$ denotes the distance between $u$ and $v$ in $S$ and $G$ respectively. In other words, string graphs are quasi-isometric to planar graphs. This theorem lifts a number of theorems from planar graphs to string graphs, we give some examples. String graphs have Assouad-Nagata (and asymptotic dimension) at most 2. Connected, locally finite, quasi-transitive string graphs are accessible. A finitely generated group $Γ$ is virtually a free product of free and surface groups if and only if $Γ$ is quasi-isometric to a string graph. Two further corollaries are that countable planar metric graphs and complete Riemannian planes are also quasi-isometric to planar graphs, which answers a question of Georgakopoulos and Papasoglu. For finite string graphs and planar metric graphs, our proofs yield polynomial time (for string graphs, this is in terms of the size of a representation given in the input) algorithms for generating such quasi-isometric planar graphs.

Figures (8)

• Figure 1: The graph $F_6$.
• Figure 2: An illustration of the resulting collection of disjoint connected sets $\mathcal{M}$. The connected sets in $\mathcal{H}$ are the red sets and the sets in $\mathcal{M}$ are the blue ones. The dashed purple lines indicate the partition of $\mathcal{M}$ into $\bigcup_{i=0}^\infty \mathcal{M}_i$.
• Figure 3: An illustration of the constructed connected sets $L^*$ and $R^^*$ (coloured purple) in a case where they do not intersect. Both $B_1$ and $B_2$ are coloured blue. The connected sets in $\mathcal{H}$ are coloured red or orange, with the orange ones being $H_\ell, H_\ell', H_r,H_r'$. The two orange paths indicate $L$ and $R$.
• Figure 4: An illustration of some $H\in \mathcal{H}$ (coloured red) that is 7-caged by $\mathcal{B}$ (coloured blue). For $u,v\in H$, the vertex sets $A_0,\ldots , A_7$ (coloured green) are as in the definition of being 7-caged. As an example illustrating the end of the proof of \ref{['lem:encase2']}, we would have that $A_0=A_u$, $A_1\in \mathcal{F}_u$, $A_2,A_5\in \mathcal{A}_{u,v}$, $\{A_3,A_4\}=\mathcal{V}_{(X,Y)}$, $A_6\in \mathcal{F}_v$, and $A_7=A_v$.
• Figure 5: A choice of $V^1_{(X,Y)},V^2_{(X,Y)}$ (coloured purple) as in the proof of \ref{['lem:encase2']}. Elements of $\mathcal{B}$ are coloured blue and elements of $\mathcal{H}$ are coloured red.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Lemma 2.2