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In this work we consider arc criticality in colourings of oriented graphs. We study deeply critical oriented graphs, those graphs for which the removal of any arc results in a decrease of the oriented chromatic number by 2. We prove the existence of deeply critical oriented cliques of every odd order , closing an open question posed by Borodin et al. Additionally, we prove the nonexistence of...
For a set $$\mathcal {H}$$ H of graphs, the $$\mathcal {H}$$ H -free Edge Deletion problem is to decide whether there exist at most k edges in the input graph, for some $$k\in \mathbb {N}$$ k ∈ N , whose deletion results in a graph without an induced copy of any of the graphs in $$\mathcal {H}$$ H . The problem is known to be fixed-parameter tractable if $$\mathcal...
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