Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond)

TL;DR

The paper studies reduced bandwidth, a qualitative strengthening of twin-width, across minor-closed graph classes and beyond. It develops distance-profile tools and product-structure methods to construct reduction sequences that bound red graphs, with applications to graph powers, map graphs, and surface-embedded graphs. Key contributions include proving bounded reduced bandwidth for all proper minor-closed classes, providing explicit planar and genus bounds (e.g., planar graphs: $\text{bw}^{\downarrow}\le 466$, $\text{tww}\le 583$), and establishing power-graph and subdivision results, alongside a separation from twin-width via expander families. The work also outlines a program of parameter-ties and open problems, highlighting limitations and guiding future investigations into dense graph classes.

Abstract

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomassé and Watrigant [J. ACM 2022] defined the twin-width of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has maximum degree at most $k$. For any graph parameter $f$, we define the reduced $f$ of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has $f$ at most $k$. Our focus is on graph classes with bounded reduced bandwidth, which implies and is stronger than bounded twin-width (reduced maximum degree). We show that every proper minor-closed class has bounded reduced bandwidth, which is qualitatively stronger than an analogous result of Bonnet et al.\ for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least $2^{1000}$. We show that planar graphs have reduced bandwidth at most $466$ and twin-width at most $583$. Our bounds for graphs of Euler genus $γ$ are $O(γ)$. Lastly, we show that fixed powers of graphs in a proper minor-closed class have bounded reduced bandwidth (irrespective of the degree of the vertices). In particular, we show that map graphs of Euler genus $γ$ have reduced bandwidth $O(γ^4)$. Lastly, we separate twin-width and reduced bandwidth by showing that any infinite class of expanders excluding a fixed complete bipartite subgraph has unbounded reduced bandwidth, while there are bounded-degree expanders with twin-width at most 6.

Figures (5)

• Figure 1: A rooted separation $(C,D)$ in a rooted tree-decomposition with root bag $B_r$.
• Figure 2: (a) The sets $X,Y^1_1,\dots,Y^1_p,Y^2_1,\dots,Y^2_q,Y^3_1,\dots,Y^3_r,Z$ in $G$. (b) The graph $H$.
• Figure 3: An illustration of $S_{x,q}^*$.
• Figure 4: The graph $S_{4,2,3}$.
• Figure 5: Identifications on $\bigcup_{j\in \{1, 2, \ldots, \ell\}} ((Q\cup Q')\setminus B)\times \{w_j\}$, when $r=1$.

Theorems & Definitions (56)

• Theorem 1: DJMMUW20UWY22
• Theorem 2: DJMMUW20UWY22
• Theorem 3: DJMMUW20
• Theorem 4: DMW23
• Lemma 5
• proof
• Corollary 6
• Lemma 7
• Lemma 8
• proof