On oriented relative clique number

An oriented graph is a directed graph with no cycle of length one or two. The relative clique number of an oriented graph is the order of a largest subset X of vertices such that each pair of vertices are either adjacent or connected by a directed 2-path. It is known that the oriented relative clique number of a planar graph is at most 80. Here we improve the upper bound to 50. We also prove an upper bound of 26 for oriented relative clique number of triangle-free planar graphs. Furthermore, we determine the exact values of oriented relative clique number of the families of outerplanar graphs with girth at least g and planar graphs with girth at least g+2 for all g≥3.