Hitting all Maximal Independent Sets of a Bipartite Graph

We prove that given a bipartite graph G with vertex set V and an integer k, deciding whether there exists a subset of V of size k hitting all maximal independent sets of G is complete for the class Sigma_2^P.Comment: v3: minor chang

Upper clique transversals in graphs

A clique transversal in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the study of the "upper" variant of this parameter, the upper clique transversal number, defined as the maximum size of a minimal clique transversal. We investigate this parameter from the algorithmic and complexity points of view, with a focus on various graph classes. We show that the corresponding decision problem is NP-complete in the classes of chordal graphs, chordal bipartite graphs, and line graphs of bipartite graphs, but solvable in linear time in the classes of split graphs and proper interval graphs.Comment: Full version of a WG 2023 pape

Dually conformal hypergraphs

Given a hypergraph $\mathcal{H}$, the dual hypergraph of $\mathcal{H}$ is the hypergraph of all minimal transversals of $\mathcal{H}$. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, etc. In this paper we study conformality of dual hypergraphs. While we do not settle the computational complexity status of recognizing this property, we show that the problem is in co-NP and can be solved in polynomial time for hypergraphs of bounded dimension. In the special case of dimension $3$, we reduce the problem to $2$-Satisfiability. Our approach has an implication in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most $k$, for any fixed $k$

Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition

Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators

Towards Exact Structural Thresholds for Parameterized Complexity

Parameterized complexity seeks to optimally use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth, i.e., graphs decomposable by small separators: Many problems admit fast algorithms relative to treewidth and many of them are optimal under the Strong Exponential-Time Hypothesis (SETH). Fewer such results are known for more general structure such as low clique-width (decomposition by large and dense but structured separators) and more restrictive structure such as low deletion distance to some sparse graph class. Despite these successes, such results remain "islands" within the realm of possible structure. Rather than adding more islands, we seek to determine the transitions between them, that is, we aim for structural thresholds where the complexity increases as input structure becomes more general. Going from deletion distance to treewidth, is a single deletion set to a graph with simple components enough to yield the same lower bound as for treewidth or does it take many disjoint separators? Going from treewidth to clique-width, how much more density entails the same complexity as clique-width? Conversely, what is the most restrictive structure that yields the same lower bound? For treewidth, we obtain both refined and new lower bounds that apply already to graphs with a single separator X such that G-X has treewidth at most r = ?(1), while G has treewidth |X|+?(1). We rule out algorithms running in time ?^*((r+1-?)^k) for Deletion to r-Colorable parameterized by k = |X|; this implies the same lower bound relative to treedepth and (hence) also to treewidth. It specializes to ?^*((3-?)^k) for Odd Cycle Transversal where tw(G-X) ? r = 2 is best possible. For clique-width, an extended version of the above reduction rules out time ?^*((4-?)^k), where X is allowed to be a possibly large separator consisting of k (true) twinclasses, while the treewidth of G - X remains r; this is proved also for the more general Deletion to r-Colorable and it implies the same lower bound relative to clique-width. Further results complement what is known for Vertex Cover, Dominating Set and Maximum Cut. All lower bounds are matched by existing and newly designed algorithms