Packing and Covering Immersions in 4-Edge-Connected Graphs

Abstract

A graph G contains another graph H as an immersion if H can be obtained from a subgraph of G by splitting off edges and removing isolated vertices. In this paper, we prove an edge-variant of the Erdős-Pósa property with respect to the immersion containment in 4-edge-connected graphs. More precisely, we prove that for every graph H, there exists a function f such that for every 4-edge-connected graph G, either G contains k pairwise edge-disjoint subgraphs where each containing H as an immersion, or there exists a set of at most f (k) edges of G intersecting all such subgraphs. This theorem is best possible in the sense that the 4-edge-connectivity cannot be replaced by the 3-edge-connectivity.

On objects dual to tree-cut decompositions

“…These ideas were later used by Dvořák and Wollan [DW16] to give a similar result for the setting of excluding strong immersions; this is a di erent, but closely related embedding notion. The connections between tree-cut decompositions and wall immersions also turned out to be important in the proof of Liu of the Erdős-Pósa property for immersion models in 4-edgeconnected graphs [Liu15]. On the algorithmic side, Giannopoulou et al [GPR + 21] used the excluded wall immersion theorem of Wollan [Wol15] and the combinatorics of tree-cut decompositions to give linear kernels for edge deletion problems for immersion-closed classes that exclude at least one subcubic planar graph as an immersion.…”

“…• G has a bramble of order k; and • G has a tangle of order k. We remark that in the last point, we mean (appropriately de ned) tangles of edge separations: partitions of the vertex set, where the edges crossing the partition form the cutset. Tangles of this avor were already considered: Liu [Liu15] used them extensively in the setting of the Erdős-Pósa property for immersion models, while Diestel and Oum [DO19] gave a suitable tangle duality theorem for the parameter carving-width, which is related to tree-cut width.…”

A global decomposition theorem for excluding immersions in graphs with no edge-cut of order three

“…A very clean local version of a decomposition theorem for excluding any fixed graph as an immersion in 4-edge-connected graphs has been obtained by the author and successfully applied to prove an Erdős-Pósa type result for packing and covering immersions [17]. As the minor relation and immersion relation are equivalent for subcubic graphs, it can be shown that once edge-cuts 2 of order 3 are allowed, any sufficiently informative decomposition theorem with respect to immersion must be at least as complicated as Robertson and Seymour's decomposition theorem with respect to minors [28], so the complicated notion of nearly embedding is unavoidable.…”

“…As Theorem 1.6 shows that the class C mentioned in Theorem 1.6 is small and addable, all conclusions of Theorem 1.7 apply to C. Now we discuss the proof of Theorem 1.4. Though the global decomposition theorem for excluding minors can be easily derived from the local version [28], it is unclear how to use similar arguments to derive Theorem 1.4 from the local decomposition in [17]. So we use a strategy different from [28] to derive Theorem 1.4 from the results in [17].…”

“…Though the global decomposition theorem for excluding minors can be easily derived from the local version [28], it is unclear how to use similar arguments to derive Theorem 1.4 from the local decomposition in [17]. So we use a strategy different from [28] to derive Theorem 1.4 from the results in [17]. A byproduct of our proof is a duality theorem for maximum order of edge-tangles and tree-cut torso-width.…”