Product structure of graph classes with bounded treewidth

Abstract

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class G to be the minimum non-negative integer c such that, for some function f , for every graph G ∈ G there is a graphWe introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of K s,t -minor-free graphs has underlying treewidth s (for t max{s, 3}); and the class of K t -minor-free graphs has underlying treewidth t − 2. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no H subgraph has bounded underlying treewidth if and only if every component of H is a subdivided star, and that the class of graphs with no induced H subgraph has bounded underlying treewidth if and only if every component of H is a star.

Product structure of graph classes with bounded treewidth

“…Bounded tree-partition-width implies bounded treewidth, as noted by Seese [69]. This fact easily generalises for c-tree-partition-width; see [12] for a proof. Of course, tw(T) = tpw(T) = 1 for every tree T. But in general, tpw(G) can be much larger than tw(G).…”

“…It is an open problem to determine the underlying treewidth of a given minor-closed class G. It is possible that the clustered chromatic number of G equals the underlying treewidth of G plus 1. This is true for any minor-closed class with underlying treewidth at most 1 by results of Norin, Scott, Seymour, and Wood [58] and Ding and Oporowski [20]; see [12]. See [58] for a conjectured value of the clustered chromatic number of G. It follows from a result of DeVos, Ding, Oporowski, Sanders, Reed, Seymour, and Vertigan [15] that every minor-closed graph class G with underlying treewidth c has clustered chromatic number at most 2(c + 1); see [12].…”

“…The definition of underlying treewidth suggests an underlying version of any graph parameter. An extended version of this paper [12] explores this idea, focusing on underlying chromatic number. It also includes details of some straightforward proofs omitted from this version.…”

“…The proof of Theorem 19 is easily adapted to show that for s 3 and fixed t, every K s,ttopological-minor-free graph G of treewidth k has s-tree-partition-width O(k 2 ); see [12]. So the underlying treewidth of the class of K s,t -topological minor-free graphs equals s if s 3 and t max{s, 3}.…”

“…Determining the underlying treewidth of the class of K s,t -topological-minor-free graphs is an interesting open problem. Also, in the s = 2 case we can use Menger's Theorem instead of Lemma 16 to show that every K 2,t -topologicalminor-free graph of treewidth k has 2-tree-partition-width O(t 2 k); see [12]. A result of Leaf and Seymour [52, (4.4)] implies that every K 2,t -minor-free graph has treewidth at most 3t/2.…”