#### Title

Detour Trees

#### Document Type

Article

#### Publication Title

Discrete Applied Mathematics

#### Publication Date

2016

#### School

College of Arts and Sciences

#### Department

Department of Mathematics

#### Abstract

A *detour* of a graph is a path of maximum length. A vertex that is common to all detours of a graph is called a *Gallai vertex*. As a tool to prove the existence of a Gallai vertex, we introduce the concept of a *detour tree*, a spanning tree of a graph in which the vertex set of any detour of the graph induces a subtree. We give several characterizations of graphs that have a detour tree. We also prove that any compatible tree of a connected dually chordal graph is a detour tree. This, combined with the fact that subtrees of a tree satisfy the Helly property, guarantees that every connected dually chordal graph contains at least one Gallai vertex. Consequently, connected graphs from subfamilies of dually chordal graphs have a Gallai vertex, including the well-studied doubly chordal, strongly chordal and interval graphs. Separately we prove that connected cographs (which are not necessarily dually chordal) have a Gallai vertex. Analogous results for cycles of maximum length follow.

#### Recommended Citation

White et al., Susan, "Detour Trees" (2016). *Mathematics Faculty Publications and Presentations*. 1.

https://scholarworks.bellarmine.edu/math_fac_pubs/1