A note on the triameter of graphs

TL;DR

This paper explores properties of the triameter in various graph classes, providing bounds, structural insights, and open problems related to triametral triples, diametral pairs, and peripheral vertices.

Contribution

It offers a tight lower bound for the triameter of trees, characterizes triametral triples in block graphs, and discusses open problems in median and distance-hereditary graphs.

Findings

Established a tight lower bound for trees' triameter

Proved that in block graphs, triametral triples contain diametral pairs

Identified open problems in median and distance-hereditary graphs

Abstract

In this note, we give answers to three questions from the paper [A. Das, Triameter of graphs, Discuss. Math. Graph Theory, 41 (2021), 601--616]. Namely, we obtain a tight lower bound for the triameter of trees in terms of order and number of leaves. We show that in a connected block graph any triametral triple of vertices contains a diametral pair and that any diametral pair of vertices can be extended to a triametral triple. We also present several open problems concerning the interplay between triametral triples, diametral pairs and peripheral vertices in median and distance-hereditary graphs.