Nowhere 0 mod p dominating sets in multigraphs

Let G be a graph with integral edge weights. A function d : V (G) → Zp is called a nowhere 0 mod p domination function if each v ∈ V satisfies ( d(v) + ∑ u∈N(v) w(u, v)d(u) ) 6= 0 mod p, where w(u, v) denotes the weight of the edge (u, v) and N(v) is the neighborhood of v. The subset of vertices with d(v) 6= 0 is called a nowhere 0 mod p dominating set. It is known that every graph has a nowhere 0 mod 2 dominating set. It is known to be false for all other primes p. The problem is open for all odd p in case all weights are one. In this paper we prove that every unicyclic graph (a graph containing at most one cycle) has a nowhere 0 mod p dominating set for all p > 1. In fact, for trees and cycles with any integral edge weights, or for any other unicyclic graph with no edge weight of (−1) mod p, there is a nowhere 0 mod p domination function d taking only 0− 1 values. This is the first nontrivial infinite family of graphs for which this property is established. We also determine the minimal graphs for which there does not exist a 0 mod p dominating set for all p > 1 in both the general case and the 0− 1 case.