Downhill and Uphill Domination in Graphs

Placing degree constraints on the vertices of a path yields the definitions of uphill and downhill paths. Specifically, we say that a path \( \pi = v_1, v_2, \ldots, v_{k+1} \) is a downhill path if for every \( i \), \( 1 \leq i \leq k \), \( \deg(v_i) \geq \deg(v_{i+1}) \). Conversely, a path \( \pi = u_1, u_2, \ldots, u_{k+1} \) is an uphill path if for every \( i \), \( 1 \leq i \leq k \), \( \deg(u_i) \leq \deg(u_{i+1}) \). 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.