Structure and algorithms for graphs excluding grids with small parity breaks as odd-minors

Abstract

We investigate a structural generalisation of treewidth we call $\mathcal{A}$-blind-treewidth where $\mathcal{A}$ denotes an annotated graph class. This width parameter is defined by evaluating only the size of those bags $B$ of tree-decompositions for a graph $G$ where ${(G,B) \notin \mathcal{A}}$. For the two cases where $\mathcal{A}$ is (i) the class $\mathcal{B}$ of all pairs ${(G,X)}$ such that no odd cycle in $G$ contains more than one vertex of ${X \subseteq V(G)}$ and (ii) the class $\mathcal{B}$ together with the class $\mathcal{P}$ of all pairs ${(G,X)}$ such that the "torso" of $X$ in $G$ is planar. For both classes, $\mathcal{B}$ and ${\mathcal{B} \cup \mathcal{P}}$, we obtain analogues of the Grid Theorem by Robertson and Seymour and FPT-algorithms that either compute decompositions of small width or correctly determine that the width of a given graph is large. Moreover, we present FPT-algorithms for Maximum Independent Set on graphs of bounded $\mathcal{B}$-blind-treewidth and Maximum Cut on graphs of bounded ${(\mathcal{B}\cup\mathcal{P})}$-blind-treewidth.

Figures (6)

• Figure 1.1: The single-crossing grid $\mathscr{U}_3$ of order $3$. The single-crossing grid $\mathscr{U}_k$ is obtained from the $(2k \times 2k)$-grid by adding a crossing to the centremost face.
• Figure 1.2: The singly parity-breaking $\mathscr{S}_3$. The red edge is the sole edge joining two vertices from the same colour class of a proper $2$-colouring of the underlying grid, hence, every odd cycle has to contain that edge.
• Figure 1.3: The three singly parity-crossing grids $\mathscr{C}^1_3$, $\mathscr{C}^2_3$, and $\mathscr{C}^3_3$ (from left to right). The red edges are those joining two vertices of the same colour class in proper $2$-colouring of the underlying grid.
• Figure 5.1: The process of turning a a $6$-wall within a $14$-wall inside out. If every drawn edge stands for a path of even length, cleanness is preserved.
• Figure 6.1: Splitting up a wall into its four quadrants. Using the dashed paths, we can extend the crossing ears of $W$ to crossing ears of one subwalls $W_i$ while collecting the vertices $x$ and $y$, marked by triangles.

Theorems & Definitions (64)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1: Tree-decomposition
• Proposition 2.2: Bodlaender1996
• Proposition 2.3: follows from Schmidt2013
• Proposition 2.4: Grohe2016
• Definition 2.5: Graph parameter