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Abstract: One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for $f(H)$ can be obtained by considering the maximum integer $k$ such that $H$ contains $k$ vertex-disjoint cycles. There exists a graph of treewidth $\Omega(k\log k)$ which does not contain $k$ vertex-disjoint cycles, from which it follows that $f(H) = \Omega(k\log k)$. In particular, if $f(H)$ is linear in $\lvert V(H) \rvert$ for graphs $H$ from a subclass of planar graphs, it is necessary that $n$-vertex graphs from the class contain at most $\lvert V(H) \rvert$ vertex-disjoint cycles. We ask whether this is also a sufficient condition, and demonstrate that this is true for classes of planar graphs with bounded component size. For an $n$-vertex graph $H$ which is a disjoint union of $r$ cycles, we show that ${f(H) \leq 3n/2 + O(r^2 \log r)}$, and improve this to $f(H)$≤$n$+O(√$n$) when $r$=2. In particular this bound is linear when $r$=O(√$n$/logn). We present a linear bound for $f(H)$ when $H$ is a subdivision of an $r$-edge planar graph for any constant~$r$. We also improve the best known bounds for $f(H)$ when $H$ is the wheel graph or the 4×4 grid, obtaining a bound of 160 for the latter.
Keywords: graph minor, treewidth, cycle packing
Published in RUP: 05.01.2026; Views: 249; Downloads: 2
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Abstract: A graph H is an induced minor of G if there exists an induced minor model of H in G, that is, a collection of pairwise disjoint subsets of vertices of G labeled by the vertices of H, each inducing a connected subgraph in G, such that two vertices of H are adjacent if and only if there is an edge in G between the corresponding subsets. In this paper, we investigate structural properties of induced minor models, including bounds on treewidth and chromatic number of the subgraphs induced by minimal induced minor models. As algorithmic applications of our structural results, we make use of recent developments regarding tree-independence number to show that if H is the 4-wheel, the 5-vertex complete graph minus an edge, or a complete bipartite graph K2,q , then there is a polynomial-time algorithm to find in a given graph G an induced minor model of H in G, if there is one. We also develop an alternative polynomial-time algorithm for recognizing graphs that do not contain K2,3 as an induced minor, which revolves around the idea of detecting the induced subgraphs whose presence is forced when the input graph contains K2,3 as an induced minor. It turns out that all these induced subgraphs are Truemper configurations.
Keywords: induced minor, treewidth, chromatic number, tree-independence number, Truemper configuration
Published in RUP: 16.12.2025; Views: 191; Downloads: 3
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Keywords: tree decomposition, treewidth, tree-independence number, induced minor, H-free graph, series parallel graph
Published in RUP: 20.09.2022; Views: 2442; Downloads: 34
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Keywords: graph class, treewidth, clique number
Published in RUP: 10.11.2020; Views: 2936; Downloads: 40
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