Detour-saturated graphs of small girths

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.