10.46298/dmtcs.6108 Grüttemeier, Niels Niels Grüttemeier Komusiewicz, Christian Christian Komusiewicz Schestag, Jannik Jannik Schestag Sommer, Frank Frank Sommer Destroying Bicolored $P_3$s by Deleting Few Edges episciences.org 2021 Computer Science - Data Structures and Algorithms Computer Science - Discrete Mathematics Mathematics - Combinatorics contact@episciences.org episciences.org 2020-02-17T10:32:22+01:00 2021-08-23T23:08:42+02:00 2021-06-08 eng Journal article https://dmtcs.episciences.org/6108 arXiv:1901.03627 1365-8050 PDF 1 Discrete Mathematics & Theoretical Computer Science ; vol. 23 no. 1 ; Graph Theory ; 1365-8050 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