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 $E_r$ 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

Destroying Multicolored Paths and Cycles in Edge-Colored Graphs

“…We first prove the NP-hardness of cP ℓ D and cC ℓ D. In our analysis we focus on the impact of cascading effects on the complexity: The NP-hardness of 2P 3 D relies on the fact that edge deletions may create new bicolored P 3 s, as 2P 3 D is polynomial-time solvable on graphs that do not provide this cascading effect [17]. Here, we show that cP ℓ D with ℓ ≥ 4 and c ∈ [2, ℓ − 2] is NP-hard even when limited to instances where G is non-cascading, defined as follows.…”

“…Proof. We prove this theorem by giving a polynomial time reduction from the NP-hard 2P 3 D problem [17].…”

“…An input graph G of an instance (G, k) of cP ℓ D (cC ℓ D) is strictly non-cascading if each subgraph of G that is a c-colored path (a c-colored cycle) is an induced subgraph. Using these terms, 2P 3 D is polynomial-time solvable on strictly non-cascading graphs [17]. We show that cP ℓ D is NP-hard on strictly non-cascading graphs for each ℓ ≥ 4 and each c ∈ [2, ℓ − 2] and that cC ℓ D is NP-hard on strictly non-cascading graphs for each ℓ ≥ 3 and each c ∈ [ℓ].…”

“…In fact, it is known that H Deletion is NP-complete for any graph H if and only if H has at least two edges [2]. Recently, it was shown that 2P 3 D is NP-hard as well [17]. If the input is restricted to graphs G where each bicolored P 3 of G is an induced subgraph of G [17], 2P 3 D is polynomial-time solvable.…”

“…Recently, it was shown that 2P 3 D is NP-hard as well [17]. If the input is restricted to graphs G where each bicolored P 3 of G is an induced subgraph of G [17], 2P 3 D is polynomial-time solvable. Furthermore, the problem of destroying not only induced, but also non-induced bicolored paths containing three vertices is polynomial-time solvable on bicolored graphs but NP-hard on 3colored graphs [9].…”