76 research outputs found
Blockers and transversals
Given an undirected graph G=(V,E) with matching number \nu(G), we define d- blockers as subsets of edges B such that \nu(G=(V,E\B))\leq \nu(G)-d. We define d- transversals T as subsets of edges such that every maximum matching M has |M\cap T|\geq d. We explore connections between d-blockers and d-transversals. Special classes of graphs are examined which include complete graphs, regular bipartite graphs, chains and cycles and we construct minimum d-transversals and d-blockers in these special graphs. We also study the complexity status of finding minimum transversals and blockers in arbitrary graphs
On Blockers and Transversals of Maximum Independent Sets in Co-Comparability Graphs
In this paper, we consider the following two problems: (i) Deletion
Blocker() where we are given an undirected graph and two
integers and ask whether there exists a subset of vertices
with such that , that
is the independence number of decreases by at least after having
removed the vertices from ; (ii) Transversal() where we are given an
undirected graph and two integers and ask whether there
exists a subset of vertices with such that for every
maximum independent set we have . We show that both
problems are polynomial-time solvable in the class of co-comparability graphs
by reducing them to the well-known Vertex Cut problem. Our results generalize a
result of [Chang et al., Maximum clique transversals, Lecture Notes in Computer
Science 2204, pp. 32-43, WG 2001] and a recent result of [Hoang et al.,
Assistance and interdiction problems on interval graphs, Discrete Applied
Mathematics 340, pp. 153-170, 2023]
Dually conformal hypergraphs
Given a hypergraph , the dual hypergraph of is the
hypergraph of all minimal transversals of . 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 , we reduce the problem to
-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 , for any
fixed
Reducing Graph Transversals via Edge Contractions
For a graph parameter ?, the Contraction(?) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which ? has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where ? is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ? according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ?, which in particular imply that Contraction(?) is co-NP-hard even for fixed k = d = 1 when ? is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when ? is the size of a minimum vertex cover, the problem is in XP parameterized by d
Contraction blockers for graphs with forbidden induced paths.
We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pâ-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs
Blockers and transversals in some subclasses of bipartite graphs: When caterpillars are dancing on a grid
Given an undirected graph G=(V,E) with matching number \nu(G), a d-blocker is a subset of edges B such that \nu(/V,E\B))= d. While the associated decision problem is NP-complete in bipartite graphs we show how to construct efficiently minimum d-transversals and minimum d-blockers in the special cases where G is a grid graph or a tree
- âŠ