Degree constraints on the vertices of a path allow for the definitions of uphill and downhill paths. Specifically, we say that a path P = vi, v2,⋯ vk+1 is a downhill path if for every i, 1 ≤ i ≤ k, deg(vi) ≥ deg(v1+1). Conversely, a path π = u1, u2,⋯ uk+1 is an uphill path if for every i, 1 ≤ i ≤ k, deg(ui) ≤ deg(ui+1). The downhill domination number of a graph G is the minimum cardinality of a set S of vertices such that every vertex in V lies on a downhill path from some vertex in S. The uphill domination number is defined as expected. We give a polynomial time algorithm to find a minimum downhill dominating set and a minimum uphill dominating set for any graph.

Placing degree constraints on the vertices of a path yields the definitions of uphill and downhill paths. Specifically, we say that a path π = v1, v2, ⋯ vk+1 is a downhill path if for every i, 1 ≤ i ≤ k, deg(v1) ≥ deg(vi+1). Conversely, a path π = u1, u2, ⋯ uk+1 is an uphill path if for every i, 1 ≤ i ≤ k, deg(u1) ≤ deg(ui+1). The downhill domination number of a graph G is defined to be the minimum cardinality of a set S of vertices such that every vertex in V lies on a downhill path from some vertex in S. The uphill domination number is defined as expected. We explore the properties of these invariants and their relationships with other invariants. We also determine a Vizing-like result for the downhill (respectively, uphill) domination numbers of Cartesian products.