The enumeration of minimal connected dominating sets is known to be notoriously hard for general graphs. Currently, it is only known that the sets can be enumerated slightly faster than $\mathcal{O}^{*}(2^n)$ and the algorithm is highly nontrivial. Moreover, it seems that it is hard to use bipartiteness as a structural aide when constructing enumeration algorithms. Hence, to the best of our knowledge, there is no known input-sensitive algorithm for enumerating minimal dominating sets, or one of their related sets, in bipartite graphs better than that of general graphs. In this paper, we provide the first input-sensitive enumeration algorithm for some non trivial subclass of bipartite graphs, namely the convex graphs. We present an algorithm to enumerate all minimal connected dominating sets of convex bipartite graphs in time $\mathcal{O}(1.7254^{n})$ where $n$ is the number of vertices of the input graph. Our algorithm implies a corresponding upper bound for the number of minimal connected dominating sets for this graph class. We complement the result by providing a convex bipartite graph, which have at least $3^{(n-2)/3}$ minimal connected dominating sets.
For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m^4 |L|^2) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m^4 |L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set with incremental-polynomial delay O(m^5 |L|) and O(m^4 |L|) respectively, when L is the set of already generated minimal edge dominating sets. Our algorithms are tailored for and solve the equivale…
It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for $K_t$-free graphs and variants. This answers a question of Kanté et al. about enumeration in bipartite graphs.
A dominating set $D$ in a graph is a subset of its vertex set such that each vertex is either in $D$ or has a neighbour in $D$. In this paper, we are interested in the enumeration of (inclusion-wise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. Firstly we show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time alg…
In the Red-Blue Dominating Set problem, we are given a bipartite graph $G = (V_B \cup V_R,E)$ and an integer $k$, and asked whether $G$ has a subset $D \subseteq V_B$ of at most $k$ "blue" vertices such that each "red" vertex from $V_R$ is adjacent to a vertex in $D$. We provide the first explicit linear kernel for this problem on planar graphs, of size at most $43k$.
A hypergraph is said to be $1$-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of $1$-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of $1$-Spernerness, thresholdness, and $2$-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of $1$-Sperner hypergraphs to …
We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330-338) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into $k$ independent dominating sets is NP-complete for every $k \geq 3$. We also strengthen a result by Henning et al. (Discrete Math. (2009), 6451-6458) …
Let G=(V,E) be a connected graph. A set U subseteq V is convex if G[U] is connected and all vertices of V\U have at most one neighbor in U. Let sigma(W) denote the unique smallest convex set that contains W subseteq V. Two players play the following game. Consider a convex set U and call it the `playground.' Initially, U = emptyset. When U=V, the player to move loses the game. Otherwise, that player chooses a vertex x in V\U which is at distance at most two from U. The effect of the move is that the playground U changes into sigma(U cup {x}) and the opponent is presented with this new playgrou…
Suppose that we are given two dominating sets $D_s$ and $D_t$ of a graph $G$ whose cardinalities are at most a given threshold $k$. Then, we are asked whether there exists a sequence of dominating sets of $G$ between $D_s$ and $D_t$ such that each dominating set in the sequence is of cardinality at most $k$ and can be obtained from the previous one by either adding or deleting exactly one vertex. This problem is known to be PSPACE-complete in general. In this paper, we study the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem remains PSP…
In this paper, we study the dualization in distributive lattices, a generalization of the well-known hypergraph dualization problem. We in particular propose equivalent formulations of the problem in terms of graphs, hypergraphs, and posets. It is known that hypergraph dualization amounts to generate all minimal transversals of a hypergraph, or all minimal dominating sets of a graph. In this new framework, a poset on vertices is given together with the input (hyper)graph, and minimal ``ideal solutions'' are to be generated. This in particular allows us to study the complexity of the problem un…
We show that the dominating set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K(1,3)), the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k^2)n^O(1) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k. Note that this parameterization of dominating set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs i…
This paper is devoted to the online dominating set problem and its variants. We believe the paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against …
The independent set on a graph $G=(V,E)$ is a subset of $V$ such that no two vertices in the subset have an edge between them. The MIS problem on $G$ seeks to identify an independent set with maximum cardinality, i.e. maximum independent set or MIS. $V* \subseteq V$ is a vertex cover $G=(V,E)$ if every edge in the graph is incident upon at least one vertex in $V*$. $V* \subseteq V$ is dominating set of $G=(V,E)$ if forall $v \in V$ either $v \in V*$ or $\exists u \in V*$ and $(u,v) \in E$. A connected dominating set, CDS, is a dominating set that forms a single component in $G$. The MVC proble…
The $k$ red domination problem for a bipartite graph $G=(X,Y,E)$ is to find a subset $D \subseteq X$ of cardinality at most $k$ that dominates vertices of $Y$. The decision version of this problem is NP-complete for general bipartite graphs but solvable in polynomial time for chordal bipartite graphs. We strengthen that result by showing that it is NP-complete for perfect elimination bipartite graphs. We present a tight upper bound on the number of such sets in bipartite graphs, and show that we can calculate that number in linear time for convex bipartite graphs. We present a linear space lin…
We prove that a fractional perfect matching in a non-bipartite graph can be written, in polynomial time, as a convex combination of perfect matchings. This extends the Birkhoff-von Neumann Theorem from bipartite to non-bipartite graphs. The algorithm of Birkhoff and von Neumann is greedy; it starts with the given fractional perfect matching and successively "removes" from it perfect matchings, with appropriate coefficients. This fails in non-bipartite graphs -- removing perfect matchings arbitrarily can lead to a graph that is non-empty but has no perfect matchings. Using odd cuts appropriatel…
Augmenting algorithms with learned predictions is a promising approach for going beyond worst-case bounds. Dinitz, Im, Lavastida, Moseley, and Vassilvitskii~(2021) have demonstrated that a warm start with learned dual solutions can improve the time complexity of the Hungarian method for weighted perfect bipartite matching. We extend and improve their framework in a principled manner via \textit{discrete convex analysis} (DCA), a discrete analog of convex analysis. We show the usefulness of our DCA-based framework by applying it to weighted perfect bipartite matching, weighted matroid intersect…
In several applications of real-time matching of demand to supply in online marketplaces, the platform allows for some latency to batch the demand and improve the efficiency. Motivated by these applications, we study the optimal trade-off between batching and inefficiency under adversarial arrival. As our base model, we consider K-stage variants of the vertex weighted b-matching in the adversarial setting, where online vertices arrive stage-wise and in K batches -- in contrast to online arrival. Our main result for this problem is an optimal (1-(1-1/K)^K)- competitive (fractional) matching alg…
