On counting limited outdegree grid digraphs and greatest increase grid digraphs

Abstract

In this paper, we introduce two special classes of digraphs. A limited outdegree grid (LOG) directed graph is a digraph derived from an $n\times n$ grid graph by removing some edges and replacing some edges with arcs such that no vertex has outdegree greater than 1. A greatest increase grid (GIG) directed graph is a LOG digraph whose vertices can be labeled with distinct labels such that each arc represents the direction of greatest increase in the underlying grid graph. We enumerate both GIG and LOG digraphs for the $3\times 3$ case.