Bounding Width on Graph Classes of Constant Diameter

TL;DR

This paper investigates how the boundedness of various graph parameters like treedepth, pathwidth, treewidth, and clique-width is affected when restricting to graphs of fixed diameter within certain classes defined by forbidden minors, induced subgraphs, or subgraphs.

Contribution

It provides classifications of when treedepth becomes bounded for graphs with fixed diameter in classes defined by forbidden subgraphs, especially focusing on $F$-subgraph-free graphs.

Findings

Classified boundedness of treedepth for diameter $d eq 2,3$ and partial results for $d=2,3$.

Identified conditions under which graph parameters transition from unbounded to bounded within restricted classes.

Focused on $F$-subgraph-free graphs for fixed diameters, providing new insights into graph parameter behavior.

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 $d=3$.