Destroying Bicolored $P_3$s by Deleting Few Edges

Abstract

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 where G contains no induced 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 $\mathcal {O}(1.85^k\cdot |V|^5)$ time and that it admits a kernel with $\mathcal {O}(\varDelta k^2)$ vertices, where $\varDelta$ is the maximum degree of G.

Some of the results of this work are contained in the third author’s Bachelor thesis [19].

F. Sommer was supported by the DFG, project MAGZ (KO 3669/4-1).