Algorithmica manuscript No.
(will be inserted by the editor)
Diameter and Treewidth in Minor-Closed Graph
Families, Revisited
Erik D. Demaine, MohammadTaghi Hajiaghayi
MIT Computer Science and Artifcial Intelligence Laboratory, 32 Vassar St., Cambridge,
MA 02139, USA.
{
edemaine,hajiagha
}
@mit.edu
The date oF receipt and acceptance will be inserted by the editor
Abstract
Eppstein [5] characterized the minor-closed graph families for which
the treewidth is bounded by a function of the diameter, which includes, e.g., planar
graphs. This characterization has been used as the basis for several (approxima-
tion) algorithms on such graphs (e.g., [2,5–8]). The proof of Eppstein is compli-
cated. In this short paper we obtain the same characterization with a simple proof.
In addition, the relation between treewidth and diameter is slightly better and ex-
plicit.
Key words
apex graphs, graph minors, bounded local treewidth, graph algo-
rithms, approximation algorithms
1 Introduction
Eppstein [5] introduced the
diameter-treewidth property
for a class of graphs,
which requires that the treewidth of a graph in the class is upper bounded by a
function of its diameter. This notion has been used extensively in a slightly mod-
iFed form called the
bounded-local-treewidth property
, which requires that the
treewidth of any connected subgraph of a graph in the class is upper bounded by
a function of its diameter. ±or minor-closed graph families, which is the focus of
most work in this context, these properties are identical.
The reason for introducing graphs of bounded local treewidth is that they have
many similar properties to both planar graphs and graphs of bounded treewidth,
two classes of graphs on which many problems are substantially easier. In partic-
ular, Baker’s approach for polynomial-time approximation schemes (PTASs) on
planar graphs [1] applies to this setting. As a result, PTASs are known for heredi-
tary maximization problems such as maximum independent set, maximum triangle
matching, maximum
H
-matching, maximum tile salvage, minimum vertex cover,