We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $ O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $ O(k\Delta\min(k,\Delta))$ vertices, where $\Delta$ is the maximum degree of $G$. Comment: 25 pages

We introduce and study the Bicolored P 3 Deletion problem defined as follows. The input is a graph G = (V, E) where the edge set E is partitioned into a set E b of blue edges and a set E r of red edges. The question is whether we can delete at most k edges such that G does not contain a bicolored P 3 as an induced subgraph. Here, a bicolored P 3 is a path on three vertices with one blue and one red edge. We show that Bicolored P 3 Deletion is NP-hard and cannot be solved in 2 o(|V |+|E|) time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored P 3 Deletion is polynomial-time solvable when G does not contain a bicolored K 3 , that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case if G contains no blue P 3 , red P 3 , blue K 3 , and red K 3 . Finally, we show that Bicolored P 3 Deletion can be solved in O(1.85 k · |V | 5 ) time and that it admits a kernel with O(∆k 2 ) vertices, where ∆ is the maximum degree of G. * Some of the results of this work are contained in the third author's Bachelor thesis [21]. † FS was supported by the DFG, project MAGZ (KO 3669/4-1).

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