Finding irrelevant vertices in linear time on bounded-genus graphs

TL;DR

A general framework that enables finding in linear time an entire set of irrelevant vertices whose removal yields a bounded-treewidth graph, provided that the input graph has bounded genus is introduced.

Abstract

The irrelevant vertex technique provides a powerful tool for the design of parameterized algorithms for a wide variety of problems on graphs. A common characteristic of these problems, permitting the application of this technique on surface-embedded graphs, is the fact that every graph of large enough treewidth contains a vertex that is irrelevant, in the sense that its removal yields an equivalent instance of the problem. The straightforward application of this technique yields algorithms with running time that is quadratic in the size of the input graph. This running time is due to the fact that it takes linear time to detect one irrelevant vertex and the total number of irrelevant vertices to be detected is linear as well. Using advanced techniques, sub-quadratic algorithms have been designed for particular problems, even in general graphs. However, designing a general framework for linear-time algorithms has been open, even for the bounded-genus case. In this paper we introduce a general framework that enables finding in linear time an entire set of irrelevant vertices whose removal yields a bounded-treewidth graph, provided that the input graph has bounded genus. Our technique consists of decomposing any surface-embedded graph into a tree-structured collection of bounded-treewidth subgraphs where detecting globally irrelevant vertices can be done locally and independently. Our method is applicable to a wide variety of known graph containment or graph modification problems where the irrelevant vertex technique applies. Examples include the (Induced) Minor Folio problem, the (Induced) Disjoint Paths problem, and the $\mathcal{F}$-Minor-Deletion problem.

Figures (4)

• Figure 1: Left: An example of a pair $(G,u_{\mathrm{root}})$ and an illustration of the bags of the radial distance decomposition of it. Vertices/faces of the same color have the same radial distance from $u_{\mathsf{root}}$ in $G$. In this example, same colored faces that share an edge are attributed to the same bag of the radial distance decomposition. Also, vertices depicted with the same color and the same style belong to the same bag of the radial distance decomposition. Right: The rooted tree $(T,t_0)$ of the radial distance decomposition of the graph on the left. Every node $t$ of $T$ is associated with a set $\chi(t)$ of vertices or faces of $G$ and is depicted as a square if $\chi(t)$ consists of faces of $G$ and as a diamond/disk/triangle if $\chi(t)$ consists of vertices of $G$. The color/style of each node $t$ of $T$ matches the color/style of the corresponding vertices and faces in $G$ that belong to $\chi(t)$. The bold edges of $T$ are its vertex-face edges.
• Figure 2: Left: Some part of the radial distance decomposition of a planar single-rooted graph. The nodes corresponding to vertex sets are depicted as orange squares and the nodes corresponding to sets of faces are depicted as blue disks. Each vertex-face edge corresponds to a same-color cycle of $G$. Right: The visualization of an embedding of ${\sf Slice}_{G}^{h}(C_e)$ in a pseudo-disk $\Psi$ where $C_e$ is the bold blue cycle on the outer boundary of $\Psi$. The cycles in $\mathsf{Cycles}(C_e,h)$ (resp. $\mathsf{Cycles}(C_e,z)$) are depicted in red (resp. blue). The black paths are disjoint paths in $G$ from the vertices of some cycle of $\mathsf{Cycles}(C_e,z)$, depicted in bold red to the vertices of $C$. As asserted by \ref{['reroute_linkage']}, all these paths can be assumed to be embedded inside $\Psi$, i.e., they do not meet the vertices of the interiors of the cycles in $\mathsf{Cycles}(C_e,h)$ (depicted in green).
• Figure 3: An example of a $\mathbb{S}^2$-embedded graph $G$ and a sequence ${\cal C}$ of 3 nested cycles in $G$. The outer/inner cycle is the one bounding the yellow/blue disk.
• Figure 4: An example of an $\mathbb{S}^2$-embedded graph $G$ (its vertices are depicted as disk-shaped nodes) and a subgraph $H$ of its radial graph $R_G$, depicted in blue. The faces of $G$ that correspond to vertices of $H$ are also drawn as yellow regions. The set $N(H)$ consists of the magenta-colored disk-shaped vertices and the grey-colored square-shaped vertices; all vertices are vertices of the radial graph $R_G$. The faces of $G$ that correspond to vertices of $H$ are also depicted as yellow regions. The faces of $G$ that are incident to vertices of $V(G)\cap V(H)$ are also depicted as cyan regions. The set $J(H)$ is set of grey-colored edges while the set $E_H(G)$ is the set of all edges of $G$ that are incident to colored faces.

Theorems & Definitions (59)

• Proposition 1
• Proposition 2
• Theorem 2.1
• Corollary 1
• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof
• Lemma 4