Detour-saturated graphs of small girths

TL;DR

This paper investigates the properties and minimal sizes of detour-saturated graphs with small girths, providing answers to longstanding open questions in graph theory.

Contribution

It answers three key questions about the existence and minimal order of detour-saturated graphs with specific girths, including the case of girth 4 and the existence beyond girth 5.

Findings

Identified the smallest detour-saturated graph of girth 4.

Confirmed the graph Pr as the smallest triangle-free detour-saturated graph.

Proved the existence of detour-saturated graphs with girth greater than 5.

Abstract

A detour of a graph G is a longest path in G. The detour order of G is the number of vertices in a detour of G. A graph is said to be detour-saturated if the addition of any edge increases strictly the detour order. L.W. Beineke, J.E. Dunbar and M. Frick asked the following three questions in 2005. (1) What is the smallest order of a detour-saturated graph of girth 4? (2) Let Pr be the graph obtained from the Petersen graph by splitting one of its vertices into three leaves. Is Pr the smallest triangle-free detour-saturated graph? (3) Does there exist a detour-saturated graph with finite girth bigger than 5? We answer these questions.