Bounding Width on Graph Classes of Constant Diameter

TL;DR

This work investigates how bounding the diameter of a graph class with initially unbounded width affects treedepth, pathwidth, treewidth, and clique-width under induced-subgraph, subgraph, and minor containment. The authors provide dichotomies and classifications, notably proving a complete treedepth dichotomy for $F$-subgraph-free graphs when the diameter $d \ge 5$ (and for $d=4$ with an additional exception) and presenting substantial partial results for $d=2,3$. They extend the analysis to the minor and induced-subgraph relations, establish diameter-based dichotomies for treewidth and clique-width, and develop intricate constructions (e.g., polarity graphs, subdivided stars, V- and E-type webs) to demonstrate boundedness or unboundedness. The findings highlight that diameter constraints can dramatically alter the width behavior of graph classes, with significant algorithmic implications via meta-theorems, and they outline several open problems and conjectures for completing the classifications in the unresolved cases.

Abstract

We determine if the width of a graph class ${\cal G}$ changes from unbounded to bounded if we consider only those graphs from ${\cal G}$ whose diameter is bounded. As parameters we consider treedepth, pathwidth, treewidth and clique-width, and as graph classes we consider classes defined by forbidding some specific graph $F$ as a minor, induced subgraph or subgraph, respectively. Our main focus is on treedepth for $F$-subgraph-free graphs of diameter at most~$d$ for some fixed integer $d$. We give classifications of boundedness of treedepth for $d\in \{4,5,\ldots\}$ and partial classifications for $d=2$ and $d=3$.

Figures (16)

• Figure 1: The subdivided "H"-graph $H_i^\ell$, the subdivided star $S_{\ell_1,\ldots, \ell_k}$, the $V$-type graph $C^V_{\ell_1,\ldots,\ell_k}$ (set of cycles sharing one common vertex) and the $E$-type graph $C^V_{\ell_1,\ldots,\ell_k}$ (set of consecutive cycles sharing an edge). We write $C^V_{\ell_1,\ldots,\ell_k}= C_{k\times [\ell_1]}^V$ and $C^E_{\ell_1,\ldots,\ell_k}=C_{k\times [\ell_1]}^E$ if $\ell_1=\dots=\ell_k$, and $C^V_{\ell_1,\ldots,\ell_k}=C_{i\times [\ell_1],k-i \times [\ell_k]}^V$ if $\ell_1=\dots=\ell_i$, $\ell_{i+1}=\dots=\ell_k$.
• Figure 2: A wall of height $2$, $3$ and $4$, respectively.
• Figure 3: The construction used in \ref{['thm:IS-CW']} to provide a $C_3$-free class of graphs with diameter $2$ and unbounded clique-width. The illustration shows a wall $W$ with a proper $2$-colouring using colours red and blue together with one additional vertex $x_i$ for every $w_i \in V(W)$.
• Figure 4: Path $Q$. Paths $Q^1, Q^2,\overline{Q}^1$ and $\overline{Q}^2$ are orange, red, blue, and green, resp.
• Figure 5: The dashed line is $P$ with vertices of $A$ in orange. Some vertices from $X(A,a)$ with their respective shortest paths are drawn, with the path $(a,a',a",q)$ is in blue.

Theorems & Definitions (46)

• theorem 1: Classification for diameter $d \geq 5$
• theorem 2: Classification for diameter $4$
• theorem 3: Partial classification for diameters $2$ and $3$
• theorem 5: GALVIN19827
• corollary 1
• theorem 10
• proof
• theorem 11
• proof
• theorem 12: Theorem 1 in Ep00