The structure of digraphs with excess one

A digraph $G$ is $k$-geodetic if for any (not necessarily distinct) vertices $u,v$ there is at most one directed walk from $u$ to $v$ with length not exceeding $k$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the directed Moore bound $M\left(d,k\right)=1+d+d^2+\cdots +d^k$. The Moore bound can be met only in the trivial cases $d=1$ and $k=1$, so it is of interest to look for $k$-geodetic digraphs with out-degree $d$ and smallest possible order $M\left(d,k\right)+ϵ$, where $ϵ$ is the excess of the digraph. Miller, Miret and Sillasen recently ruled out the existence of digraphs with excess one for $k=3,4$ and $d\ge 2$ and for $k=2$ and $d\ge 8$. We conjecture that there are no digraphs with excess one for $d,k\ge 2$ and in this paper we investigate the structure of minimal counterexamples to this conjecture. We severely constrain the possible structures of the outlier function and prove the nonexistence of certain digraphs with degree three and excess one, as well closing the open cases $k=2$ and $d=3,4,5,6,7$ left by the analysis of Miller et al. We further show that there are no involutary digraphs with excess one, that is, the outlier function of any such digraph must contain a cycle of length $\ge 3$.