On oriented cliques with respect to push operation

Julien Bensmail 1 Soumen Nandi 2 Sagnik Sen 3
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : An oriented graph is a directed graph without any directed cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. To push a vertex v of a directed graph G is to change the orientations of all the arcs incident to v. 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 the underlying graph of a push clique or not. We also prove that a planar push clique can have at most 8 vertices and provide an exhaustive list of planar push cliques.
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Julien Bensmail, Soumen Nandi, Sagnik Sen. On oriented cliques with respect to push operation. Discrete Applied Mathematics, Elsevier, 2017, 232, pp.50 - 63. ⟨10.1016/j.dam.2017.07.037⟩. ⟨hal-01629946⟩

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