Hereditary Graph Product Structure Theory and Induced Subgraphs of Strong Products

TL;DR

The paper develops a hereditary refinement of graph product structure by introducing ${\cal H}$-clique-width and proving induced-subgraph containment theorems for strong products. It shows that graphs subgraph-embeddable in $Q\boxtimes M$ with bounded degree and bounded tree-width can be realized as induced subgraphs of $Q\boxtimes M'$ or $Q\boxtimes M''$ with width-bounded factors, and it provides explicit width bounds. A key achievement is the induced planar product-structure result, giving $G$ as an induced subgraph of $P\boxtimes M$ with $tw(M)\le 39$. The framework connects to algorithmic implications via local clique-width and twin-width, and opens multiple avenues for FO logic, transductions, and broader choices of ${\cal H}$. Overall, the work integrates hereditary product structure with concrete width bounds, enabling induced-structure analyses for planar and other sparse/dense graph classes.

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).

Figures (2)

• Figure 1: Illustrating the strong product $\boxtimes$ of the two shaded graphs.
• Figure 2: A $(P,5)$-expression making an $a\times b$ square grid, where $P$ is a $b$-vertex loop path: Left: The loop path $P\in{\cal H}$ with the parameter vertices $v_1,v_2,\ldots,v_b$ (here $b=4$). Right: Making the grid with a $(P,5)$-expression, left-to-right and top-to-bottom in order (a)--(l), as follows. (a) Create two vertices labelled $(1,v_1)$ (red) and $(5,v_2)$ (blue). (b) Add edges from red to blue. (c) Recolour $(5,v_2)$ to $(2,v_2)$ (light red) and add a new vertex labelled $(5,v_3)$ (blue). (d) Add edges from light red to blue, then recolour blue and add a new vertex labelled $(5,v_4)$ (blue). (e) Add edges from red to blue again, and the recolour blue to light red, finishing a copy $P_1$ of the path $P$. (f) Analogously create a copy $P_2$ of $P$, coloured alternately green and light green. (g) Add edges from red to green -- note that this creates only the two "horizontal" edges because of $P$. (h) Analogously add edges from light red to light green. (i) Recolour both red and light red to blue, and then add a copy $P_3$ of $P$ coloured red and light red. (j) Add edges from green to red and from light green to light red, as before. (k) Analogously recolour both green and light green to blue, and then add a copy $P_4$ of $P$ coloured green and light green. (l) Continue this construction up to desired size.

Theorems & Definitions (18)

• Theorem 1: simplified \ref{['thm:fromprods']}
• Theorem 2: \ref{['cor:fromprods']}
• Theorem 3: \ref{['thm:onlyplanar']}
• Theorem 4: $\!$DBLP:journals/jacm/DujmovicJMMUW20, improved in DBLP:journals/combinatorics/UeckerdtWY22
• Definition 5: ${\cal H}$-clique-width
• Claim 6
• Theorem 7
• Lemma 9
• Theorem 10
• Corollary 11