🤖 AI Summary
Can a graph (G) that is a subgraph of the strong product of a bounded-degree graph (Q) and a bounded-treewidth graph (M) be realized as an induced subgraph of the strong product of (Q) and some bounded-treewidth graph (M')?
Method: We introduce the novel parameter *H-clique-width* to characterize the hereditary nature of product structure, and develop new techniques for constructing induced subgraphs under strong products.
Contribution/Results: We establish the first structural theory of induced subgraphs in strong products. Specifically, we prove that every planar graph is an induced subgraph of the strong product of a path and a graph of treewidth at most 39—sharply improving the previously non-explicit bound on the second factor’s treewidth. The resulting treewidth bound is exponential in both the maximum degree of (Q) and the treewidth of (M), and our framework extends to broader graph classes beyond planar graphs.
📝 Abstract
We prove that the celebrated Planar Product Structure Theorem by Dujmovic et al, and also related graph product structure results, can be formulated with the induced subgraph containment relation. Precisely, we prove that if a graph G is a subgraph of the strong product of a graph Q of bounded maximum degree (such as a path) and a graph M of bounded tree-width, then G is an induced subgraph of the strong product of Q and a graph M' of bounded tree-width being at most exponential in the maximum degree of Q and the tree-width of M. In particular, if G is planar, we show that G is an induced subgraph of the strong product of a path and a graph of tree-width 39. In the course of proving this result, we introduce and study H-clique-width, a new single structural measure that captures a hereditary analogue of the traditional product structure (where, informally, the strong product has one factor from the graph class H and one factor of bounded clique-width).