We study the parameterized and classical complexity of two problems that are concerned with induced paths on three vertices, called P 3 s, in undirected graphs G = (V, E). In Strong Triadic Closure we aim to label the edges in E as strong and weak such that at most k edges are weak and G contains no induced P 3 with two strong edges. In Cluster Deletion we aim to destroy all induced P 3 s by a minimum number of edge deletions. We first show that Strong Triadic Closure admits a 4k-vertex kernel. Then, we study parameterization by ℓ := |E| − k and show that both problems are fixed-parameter tractable and unlikely to admit a polynomial kernel with respect to ℓ. Finally, we give a dichotomy of the classical complexity of both problems on H-free graphs for all H of order at most four. IntroductionWe study two related graph problems arising in social network analysis and data clustering. Assume we are given a social network where vertices represent agents and edges represent relationships between these agents, and want to predict which of the relationships are important. In online social networks for example, one could aim to distinguish between close friends and spurious relationships. Sintos and Tsaparas [28] proposed to use the notion of strong triadic closure for this problem. This notion goes back to Granovetter's sociological work [10]. Informally, it is the assumption that if one agent has strong relationships with two other agents, then these two other agents should have at least a weak relationship. The aim in the computational problem is then to label a maximum number of edges of the given social network as strong while fulfilling this requirement. Formally, we are looking for an STC-labeling defined as follows. ⋆ A preliminary version of this work appeared inKnown Results. STC is NP-hard [28], even when restricted to graphs with maximum degree four [16] or to split graphs [17]. In contrast, STC is solvable in polynomial time when the input graph is bipartite [28], subcubic [16], a proper interval graph [17], or a cograph, that is, a graph with no induced P 4 [16]. STC can be solved in O(1.28 k + nm) time and admits a polynomial kernel when parameterized by k. These two results follow from a parameter-preserving reduction to Vertex Cover, which asks if it is possible to delete at most k vertices of a given graph such that the remaining graph does not contain any edge. This parameter-preserving reduction [28] computes the so-called Gallai graph [18,29] of the input graph and directly gives the above-mentioned running time bound. The existence of a kernel for parameter k is implied by two facts: First, Vertex Cover admits a polynomial kernel for the number k of vertices to delete [6,7]. Second, Vertex Cover is in NP and STC is NP-hard. Hence, the Vertex Cover instance of size poly(k) which we obtain by first reducing from STC to Vertex Cover and then applying the kernelization can be transformed into an equivalent STC instance by a polynomial-time reduction. The STC instance then has size poly(k).

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

We study the problem of learning the structure of an optimal Bayesian network when additional constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the network or its moralized graph are close, in terms of vertex or edge deletions, to a sparse graph class Π. For example, we show that learning an optimal network whose moralized graph has vertex deletion distance at most k from a graph with maximum degree 1 can be computed in polynomial time when k is constant. This extends previous work that gave an algorithm with such a running time for the vertex deletion distance to edgeless graphs. We then show that further extensions or improvements are presumably impossible. For example, we show that learning optimal networks where the network or its moralized graph have maximum degree 2 or connected components of size at most c, c ≥ 3, is NP-hard. Finally, we show that learning an optimal network with at most k edges in the moralized graph presumably has no f(k) · |I|O(1)-time algorithm and that, in contrast, an optimal network with at most k arcs can be computed in 2O(k) · |I|O(1) time where |I| is the total input size.

Given an undirected graph G = (V, E) the NP-hard Strong Triadic Closure (STC) problem asks for a labeling of the edges as weak and strong such that at most k edges are weak and for each induced P3 in G at least one edge is weak. In this work, we study the following generalizations of STC with c different strong edge colors. In Multi-STC an induced P3 may receive two strong labels as long as they are different. In Edge-List Multi-STC and Vertex-List Multi-STC we may additionally restrict the set of permitted colors for each edge of G. We show that, under the ETH, Edge-List Multi-STC and Vertex-List Multi-STC cannot be solved in time 2 o(|V | 2 ) , and that Multi-STC is NP-hard for every fixed c. We then proceed with a parameterized complexity analysis in which weextend previous fixed-parameter tractability results and kernelizations for STC [Golovach et al., SWAT '18, Grüttemeier and Komusiewicz, WG '18] to the three variants with multiple edge colors or outline the limits of such an extension.

We study the problem of learning the structure of an optimal Bayesian network when additional structural constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the moralized graph can be transformed to a graph from a sparse graph class Π by at most k vertex deletions. We show that for Π being the graphs with maximum degree 1, an optimal network can be computed in polynomial time when k is constant, extending previous work that gave an algorithm with such a running time for Π being the class of edgeless graphs [Korhonen & Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that when Π is the set of graphs in which each component has size at most three, then learning an optimal network is NP-hard even if k=0. Finally, we show that learning an optimal network with at most k edges in the moralized graph presumably is not fixed-parameter tractable with respect to k and that, in contrast, computing an optimal network with at most k arcs can be computed is fixed-parameter tractable in k.