Polynomial-time algorithm for total transitivity in split graphs

Design an algorithm for computing total transitivity in split graphs, extending the known linear-time algorithm for computing transitivity in that graph class.

Background

The paper establishes structural characterizations for split graphs with total transitivity equal to 1 and to one less than the clique number, and it derives necessary conditions for intermediate values. However, it does not provide an algorithm for computing total transitivity on arbitrary split graphs. The authors contrast this gap with the known linear-time algorithm for the ordinary transitivity problem in split graphs and explicitly pose the algorithmic question of whether total transitivity can likewise be computed efficiently.