Structure and algorithms for graphs excluding grids with small parity breaks as odd-minors
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912054365913088 |
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| author | Gollin, J. Pascal Wiederrecht, Sebastian |
| author_facet | Gollin, J. Pascal Wiederrecht, Sebastian |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_04504 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Structure and algorithms for graphs excluding grids with small parity breaks as odd-minors Gollin, J. Pascal Wiederrecht, Sebastian Combinatorics Discrete Mathematics 05C83, 05C85, 05C75, 68R10, 68Q27 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. |
| title | Structure and algorithms for graphs excluding grids with small parity breaks as odd-minors |
| topic | Combinatorics Discrete Mathematics 05C83, 05C85, 05C75, 68R10, 68Q27 |
| url | https://arxiv.org/abs/2304.04504 |