Convex Hulls and Simple Colourings in Directed and $2$-edge-Coloured Graphs

An oriented graph ($2$-edge-coloured graph) is complete convex when the convex hull of every arc (edge) is the entirety of the vertex set. Here we show the problem of bounding the oriented ($2$-edge-coloured) chromatic number of the family of planar graphs can be restricted to bounding this parameter for the family of planar complete-convex oriented ($2$-edge-coloured) graphs. We fully classify complete-convex oriented and $2$-edge-coloured graphs with tree-width $2$. Further we show that it is NP-complete to decide if a graph is the underlying graph of a complete-convex oriented or $2$-edge-coloured graph even when restricted to inputs with tree-width $4$.