  We study the complexity of graph modification problems with respect to
homomorphism-based colouring properties of edge-coloured graphs. A homomorphism
from edge-coloured graph $G$ to edge-coloured graph $H$ is a vertex-mapping
from $G$ to $H$ that preserves adjacencies and edge-colours. We consider the
property of having a homomorphism to a fixed edge-coloured graph $H$. The
question we are interested in is: given an edge-coloured graph $G$, can we
perform $k$ graph operations so that the resulting graph admits a homomorphism
to $H$? The operations we consider are vertex-deletion, edge-deletion and
switching (an operation that permutes the colours of the edges incident to a
given vertex). Switching plays an important role in the theory of signed
graphs, that are 2-edge-coloured graphs whose colours are the signs $+$ and
$-$. We denote the corresponding problems (parameterized by $k$) by
VD-$H$-COLOURING, ED-$H$-COLOURING and SW-$H$-COLOURING. These problems
generalise $H$-COLOURING (to decide if an input graph admits a homomorphism to
a fixed target $H$).
  Our main focus is when $H$ is an edge-coloured graph with at most two
vertices, a case that is already interesting as it includes problems such as
VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph
$H$, we give a P/NP-c complexity dichotomy for VD-$H$-COLOURING,
ED-$H$-COLOURING and SW-$H$-COLOURING. We then address their parameterized
complexity. We show that VD-$H$-COLOURING and ED-$H$-COLOURING for all such $H$
are FPT. In contrast, already for some $H$ of order 3, unless P=NP, none of the
three problems is in XP, since 3-COLOURING is NP-c. We show that
SW-$H$-COLOURING is different: there are three 2-edge-coloured graphs $H$ of
order 2 for which SW-$H$-COLOURING is W-hard, and assuming the ETH, admits no
algorithm in time $f(k)n^{o(k)}$. For the other cases, SW-$H$-COLOURING is FPT.
