We consider the Maximum 2-Club problem where one is given as input an undirected graph G = (V,E) and seeks a subset of vertices S of maximum size such that any pair of vertices in S is connected by a path of length at most 2 in the graph induced by S. This problem is a natural relaxation of the famous Maximum Clique problem where any pair of vertices must be connected by an edge. Maximum 2-Club has been well-studied and is known to be NP-complete even on split graphs. It can be solved exactly in O^*(1.62ⁿ) time, where n denotes the number of vertices of the input graph, while being polynomial-time solvable on several graph classes. Parameterized algorithms for structural parameters have also been considered, leading in particular to an algorithm with a double-exponential dependence in the parameter treewidth. Such an algorithm is actually the best one known for the larger parameter vertex cover size up to a constant in the exponent. We provide new results in both directions. We first prove that the double-exponential dependence for parameter vertex cover size is unavoidable under the Exponential Time Hypothesis (ETH). This answers a question left open by Hartung, Komusiewicz, Nichterlein and Suchỳ [Hartung et al., 2015]. Our result also implies that the problem cannot be solved in time sub-exponential in n even for split graphs. We then provide an exact algorithm for the problem restricted to chordal graphs, running in O^*(1.1996ⁿ) time, by reducing Maximum 2-Club on this class to Maximum Independent Set on arbitrary graphs with the same number of vertices. The same reduction shows that we can enumerate all maximum (and inclusion-wise maximal) 2-clubs of a chordal graph in O^*(3^{n/3}) = O^*(1.4423ⁿ) time. We conclude by providing a construction of split graphs with Ω(3^{n/3}/poly(n)) maximum2-clubs, for some polynomial poly showing that the bound for enumeration is essentially tight.

Abstract. The question to enumerate all (inclusionwise) minimal connected dominating sets in a graph of order [Formula: see text] in time significantly less than [Formula: see text] is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time [Formula: see text], using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time [Formula: see text]. Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order [Formula: see text] with [Formula: see text] many minimal connected dominating sets, while previous examples achieved [Formula: see text]. Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are [Formula: see text] and [Formula: see text], respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much effort. More precisely, we prove that it is NP -complete to decide, given a graph [Formula: see text] and a vertex set [Formula: see text], if there exists a minimal connected dominating set [Formula: see text] with [Formula: see text], even if [Formula: see text] is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT -algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by [Formula: see text]. This also adds one more problem to the still rather few natural parameterized problems that are complete for the parameterized complexity class W [3]. We also relate our enumeration problem to the famous Hitting Set Transversal problem, a problem open for more than four decades, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay, by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic (polynomial-delay) solution to the Hitting Set Transversal problem.

The question to enumerate all (inclusion-wise) minimal connected dominating sets in a graph of order n in time significantly less than 2ⁿ is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time 𝒪(1.9896ⁿ), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time 𝒪(1.9767ⁿ). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with Ω(1.4890ⁿ) many minimal connected dominating sets, while previous examples achieved Ω(1.4422ⁿ). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are Ω(1.3195ⁿ) and Ω(1.4723ⁿ), respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much efforts. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U, if there exists a minimal connected dominating set D with U ⊆ D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by |U|. This also adds one more problem to the still rather few natural parameterized problems that are complete for the class W[3]. We also relate our enumeration problem to the famous open Hitting Set Transversal problem, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic solution to the Hitting Set Transversal problem.

Inferring probabilistic networks from data is a notoriously difficult task. Under various goodness-of-fit measures, finding an optimal network is NP-hard, even if restricted to polytrees of bounded in-degree. Polynomial-time algorithms are known only for rare special cases, perhaps most notably for branchings, that is, polytrees in which the in-degree of every node is at most one. Here, we study the complexity of finding an optimal polytree that can be turned into a branching by deleting some number of arcs or nodes, treated as a parameter. We show that the problem can be solved via a matroid intersection formulation in polynomial time if the number of deleted arcs is bounded by a constant. The order of the polynomial time bound depends on this constant, hence the algorithm does not establish fixed-parameter tractability when parameterized by the number of deleted arcs. We show that a restricted version of the problem allows fixed-parameter tractability and hence scales well with the parameter. We contrast this positive result by showing that if we parameterize by the number of deleted nodes, a somewhat more powerful parameter, the problem is not fixed-parameter tractable, subject to a complexity-theoretic assumption.

The minimum dominating set problem remains NP-hard when restricted to any of the following graph classes: c -dense graphs, chordal graphs, 4-chordal graphs, weakly chordal graphs, and circle graphs. Developing and using a general approach, for each of these graph classes we present an exponential time algorithm solving the minimum dominating set problem faster than the best known algorithm for general graphs. Our algorithms have the following running time: O (1.4124 n ) for chordal graphs, O (1.4776 n ) for weakly chordal graphs, O (1.4845 n ) for 4-chordal graphs, O (1.4887 n ) for circle graphs, and O (1.2273 (1+√1−2 c ) n ) for c -dense graphs.