The Minimum Number of Dependent Arcs and a Related Parameter of Generalized Mycielski Graphs

TL;DR

This paper investigates the minimum number of dependent arcs in acyclic orientations of generalized Mycielski graphs and explores related parameters like the minimum edge deletions needed to obtain cover graphs, extending previous results.

Contribution

It generalizes existing results on dependent arcs from Mycielski graphs to their k-parameterized versions and introduces bounds for related parameters such as cover graph edge deletions.

Findings

Derived bounds for m(M_k(G)) in generalized Mycielski graphs.

Extended results on c(G) for cover graphs.

Connected dependent arc counts to graph transformations.

Abstract

Let D be an acyclic orientation of the graph G. An arc of D is dependent if its reversal creates a directed cycle. Let m(G) denote the minimum number of dependent arcs over all acyclic orientations of G. For any k > 0, a generalized Mycielski graph M_k(G) of G is defined. Note that M_1(G) is the usual Mycielskian of G. We generalize results concerning m(M_1(G)) in K. L. Collins, K. Tysdal, J. Graph Theory, 46 (2004), 285-296, to m(M_k(G)). The underlying graph of a Hasse diagram is called a cover graph. Let c(G) denote the the minimum number of edges to be deleted from a graph G to get a cover graph. Analogue results about c(G) are also obtained.