On $(1,2)$-step competition graphs of bipartite tournaments

TL;DR

This paper investigates the structure of $(1,2)$-step competition graphs derived from bipartite tournaments, revealing their possible configurations, characterizations, and extremal edge counts, with implications for graph representation.

Contribution

It provides a comprehensive structural characterization of $(1,2)$-step competition graphs of bipartite tournaments, including their components, special cases, and extremal edge bounds.

Findings

The $(1,2)$-step competition graph has at most one non-trivial component or two complete components of size at least three.

$K_{1,4}$ is the unique connected non-complete, triangle-free, or cycle-edge-disjoint graph representable as such.

Complete graphs and unions of two complete graphs are characterized as representable, with bounds on maximum and minimum edges.

Abstract

In this paper, we study $(1,2)$-step competition graphs of bipartite tournaments. A bipartite tournament means an orientation of a complete bipartite graph. We show that the $(1,2)$-step competition graph of a bipartite tournament has at most one non-trivial component or consists of exactly two complete components of size at least three and, especially in the former, the diameter of the nontrivial component is at most three if it exists. Based on this result, we show that, among the connected non-complete graphs which are triangle-free or the cycles of which are edge-disjoint, $K_{1,4}$ is the only graph that can be represented as the $(1,2)$-step competition graph of a bipartite tournament. We also completely characterize a complete graph and the disjoint union of two complete graphs, respectively, which can be represented as the $(1,2)$-step competition graph of a bipartite tournament. Finally we present the maximum number of edges and the minimum number of edges which the $(1,2)$-step competition graph of a bipartite tournament might have.