Complexity of locally-injective homomorphisms to tournaments

Given two graphs G = (VG, EG) and H = (VH , EH), a homomorphism from G to H is a function f : VG → VH such that for every uv ∈ EG, f(u)f(v) ∈ EH . A homomorphism from G to H is referred to as an H-colouring of G and the vertices of H are regarded as colours. The graph H is called the target of the homomorphism. These definitions extend to directed graphs by requiring that the mapping must preserve the existence as well as the direction of each arc. A locally-injective homomorphism f from G to H is a homomorphism from G to H such that for every v ∈ V the restriction of f to N(v) (or possibly N [v] = N(v)∪{v}) is injective. The complexity of locally-injective homomorphisms for undirected graphs has been examined by a variety of authors and in a variety of contexts [6, 7, 10, 11, 12, 13, 14, 21]. Locally-injective homomorphisms of graphs find application in a range of areas including bio-informatics [1, 8, 9] and coding theory [14]. Here we consider locally-injective homomorphisms of oriented graphs, that is, directed graphs in which any two vertices are joined by at most one arc. Given a vertex v, an arc from v to v is called a loop. A directed graph with a loop at every vertex is called reflexive; a directed graph with no loops is called irreflexive. To define locally-injective homomorphisms of oriented graphs, one must choose the neighbourhood(s) on which the homomorphism must be injective. Up to symmetry, there are four natural choices: (1) N−(v). (2) N+(v) and also N−(v). (3) N+(v) ∪N−(v). (4) N+[v] ∪N−[v] = N+(v) ∪N−(v) ∪ {v}. For irreflexive targets, (2), (3) and (4) are equivalent. Under (4), adjacent vertices must always be assigned different colours, and hence whether or not the target contains loops is irrelevant. Therefore, we may assume that targets are irreflexive when considering (4). Then, a locallyinjective homomorphism to an irreflexive target satisfying (4) is equivalent to a locally-injective homomorphism to the same irreflexive target under either (2) or (3). As such, we need not consider (4) and are left with three distinct cases. Taking (1) as our injectivity requirement defines in-injective homomorphism; taking (2) defines ios-injective homomorphism; and taking (3) defines iot-injective homomorphism. Here “ios” and “iot” stand for “in and out separately” and “in and out together” respectively. The problem of in-injective homomorphism is examined by MacGillivray, Raspaud, and Swarts in [18, 19]. They give a dichotomy theorem for the problem of in-injective homomorphism to reflexive oriented graphs; and one for the problem of in-injective homomorphism to irreflexive tournaments. The problem of in-injective homomorphism to irreflexive oriented graphs H is shown to be NP-complete when the maximum in-degree of H, ∆−(H), is at least 3, and Polynomial when ∆−(H) = 1. For the case ∆−(H) = 2 they show that an instance of directed graph homomorphism polynomially transforms to an instance of in-injective homomorphism to a target with maximum