Complexity of Injective Homomorphisms to Small Tournaments, and of Injective Oriented Colourings

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 prese...

The complexity of locally injective homomorphisms

A homomorphism f : G → H, from a digraph G to a digraph H, is locally injective if the restriction of f to N(v) is an injective mapping, for each v ∈ V (G). The problem of deciding whether such an f exists is known as the injective H-colouring problem (INJ-HOMH). In this paper we classify the problem INJ-HOMH as either being a problem in P or a problem that is NPcomplete. This is done in the ca...

On Injective Colourings of Chordal Graphs

We show that one can compute the injective chromatic number of a chordal graph G at least as efficiently as one can compute the chromatic number of (G−B), where B are the bridges of G. In particular, it follows that for strongly chordal graphs and so-called power chordal graphs the injective chromatic number can be determined in polynomial time. Moreover, for chordal graphs in general, we show ...

Cohen-macaulay F -injective Homomorphisms

Mining Tree Patterns with Partially Injective Homomorphisms

One of the main differences between ILP and graph mining is that while pattern matching in ILP is mainly defined by homomorphism (subsumption), it is the subgraph isomorphism in graph mining. Using that subgraph isomorphisms are injective homomorphisms, we bridge the gap between the two pattern matching operators with partially injective homomorphisms, which are homomorphisms requiring the inje...