Nowhere \(0\mod p\;\) Dominating Sets in Multigraphs

Abstract

Let \(G\) be a graph with integral edge weights. A function \(d: V(G) \to \mathbb{Z}_p\) is called a nowhere \(0 \mod p\) domination function if each \(v \in V\) satisfies \((\sum_{u \in N(v)} w(u,v)d(u))\neq 0 \mod p\), 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 \(d(v) \neq 0\) 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.