On oriented cliques with respect to push operation

TL;DR

This paper investigates push cliques in oriented graphs, proving NP-completeness for recognizing underlying graphs, establishing a maximum size for planar push cliques, and providing a complete classification of minimal planar push cliques.

Contribution

It introduces the concept of push cliques, proves recognition complexity, bounds their size in planar cases, and classifies minimal instances.

Findings

Recognizing underlying graphs of push cliques is NP-complete.

Maximum size of planar push cliques is 8 vertices.

Complete list of minimal planar push cliques is provided.

Abstract

To push a vertex $v$ of a directed graph $\overrightarrow{G}$ is to change the orientations of all the arcs incident with $v$. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. A push clique is an oriented clique that remains an oriented clique even if one pushes any set of vertices of it. We show that it is NP-complete to decide if an undirected graph is underlying graph of a push clique or not. We also prove that a planar push clique can have at most 8 vertices. We also provide an exhaustive list of minimal (with respect to spanning subgraph inclusion) planar push cliques.