String Graphs: Product Structure and Localised Representations

Abstract

We investigate string graphs through the lens of graph product structure theory, which describes complicated graphs as subgraphs of strong products of simpler building blocks. A graph $G$ is called a string graph if its vertices can be represented by a collection $\mathcal{C}$ of continuous curves (called a string representation of $G$) in a surface so that two vertices are adjacent in $G$ if and only if the corresponding curves in $\mathcal{C}$ cross. We prove that every string graph with bounded maximum degree in a fixed surface is isomorphic to a subgraph of the strong product of a graph with bounded treewidth and a path. This extends recent product structure theorems for string graphs. Applications of this result are presented. This product structure theorem ceases to be true if the `bounded maximum degree' assumption is relaxed to `bounded degeneracy'. For string graphs in the plane, we give an alternative proof of this result. Specifically, we show that every string graph in the plane has a `localised' string representation where the number of crossing points on the curve representing a vertex $u$ is bounded by a function of the degree of $u$. Our proof of the product structure theorem also leads to a result about the treewidth of outerstring graphs, which qualitatively extends a result of Fox and Pach [Eur. J. Comb. 2012] about outerstring graphs with bounded maximum degree. We extend our result to outerstring graphs defined in arbitrary surfaces.

Figures (5)

• Figure 1: An example of fragments and sections. (a) A string representation $\mathcal{C}$ in the plane with an ordered $5$-colouring $\phi$, where colours are: green $= 1$, blue $= 2$, black $= 3$ (only the horizontal curve $\gamma$ is black), red $= 4$, and brown $= 5$. The curve $\gamma$ is split by the crossing points (marked as squares) of $\gamma$ and the curves of smaller colours into five fragments (highlighted in purple). (b) The planarisation $\mathcal{C}'$ of $\mathcal{C}$. Each vertex is labelled by its level. The edges of sections and $1$-vertex sections of $\mathcal{C}'$ are highlighted in purple. There are three sections of $L_{\gamma}$, one of which consists of a single dummy vertex labelled $d$.
• Figure 2: The coloured planarisation $\mathcal{C}^{\phi}$ of the string representation $\mathcal{C}$ with the ordered $5$-colouring $\phi$ from \ref{['fragmentssectionsa']}. Each vertex is labelled by its level. The edges between consecutive vertices of the walk $W_{\gamma}$ in $\mathcal{C}^{\phi}$ are highlighted in purple.
• Figure 3: Proof of \ref{['main:localised']}. (a) A string representation $\mathcal{C}_{0}$. The points of $X$ are marked as black squares. For every two crossing curves, exactly one of their common crossing points is in $X$. (b) The drawing $D_{0}$ of the graph $H$ obtained from $\mathcal{C}_{0}$ and $X$. The vertices of $H$ are grey.
• Figure 4:
• Figure 5: A collection $\mathcal{C}$ of horizontal and vertical segments. The horizontal segment $\gamma$ (drawn in green) crosses $t$ vertical segments $\gamma_{1}, \dots, \gamma_{t}$ (drawn in blue) ordered 'from left to right'. For each $i \in \{1, \dots, t\}$ and $j \in \{1, \dots, t\}$, the horizontal segment $\alpha_{i}^{j}$ (drawn in red) crosses $\gamma_{i}$ and no other segments of $\{\gamma_{1}, \dots, \gamma_{t}\}$. For each $i \in \{1, \dots, t - 1\}$ and $j \in \{1, \dots, t\}$, the vertical segment $\beta_{i}^{j}$ (drawn in black) crosses two segments $\alpha_{i}^{j}$, $\alpha_{i + 1}^{j}$ and crosses no other segment of $\mathcal{C}$.

Theorems & Definitions (48)

• Theorem 1
• Theorem 2: FP-EJC12
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6: KM91
• Theorem 7: Schaefer18SS-JCSS04
• Theorem 8: SSS-JCSS03
• Theorem 9: HW24
• Theorem 9