Destroying Bicolored P_3s by Deleting Few Edges
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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(Δ k^2) vertices, where Δ is the maximum degree of G.
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