Graph Modification for Edge-Coloured and Signed Graph Homomorphism Problems: Parameterized and Classical Complexity

We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from an edge-coloured graph G to an 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, which generalises the classic vertex-colourability property. The question we are interested in is the following: 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 the extensively studied H-Colouring problem (where one has to decide if an input graph admits a homomorphism to a fixed target H). For 2-edge-coloured H, it is known that H-Colouring already captures the complexity of all fixed-target Constraint Satisfaction Problems. Our main focus is on the case where H is an edge-coloured graph with at most two vertices, a case that is already interesting since it includes standard problems such as Vertex Cover, Odd Cycle Transversal and Edge Bipartization. For such a graph H, we give a P/NP-complete complexity dichotomy for all three VD-H-Colouring, ED-H-Colouring and SW-H-Colouring problems. Then, we address their parameterized complexity. We show that all VD-H-Colouring and ED-H-Colouring problems for such H are FPT. This is in contrast with the fact that already for some H of order 3, unless P = NP, none of the three considered problems is in XP, since 3- Colouring is NP-complete. We show that the situation is different for SW-H-Colouring: there are three 2-edge-coloured graphs H of order 2 for which SW-H-Colouring is W[1]-hard, and assuming the ETH, admits no algorithm in time $f(k)n^{o(k)}$ for inputs of size n and for any computable function f. For the other cases, SW-H-Colouring is FPT.