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
Polynomial-time algorithm for Maximum Independent Set in bounded-degree graphs with no long induced claws
