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