88 research outputs found
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
Coloring translates and homothets of a convex body
We obtain improved upper bounds and new lower bounds on the chromatic number
as a linear function of the clique number, for the intersection graphs (and
their complements) of finite families of translates and homothets of a convex
body in \RR^n.Comment: 11 pages, 2 figure
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]
Approximation algorithms for combinatorial optimization problems
Fil: Vassiliev, Saveli. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina
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
Rees algebras, Monomial Subrings and Linear Optimization Problems
In this thesis we are interested in studying algebraic properties of monomial
algebras, that can be linked to combinatorial structures, such as graphs and
clutters, and to optimization problems. A goal here is to establish bridges
between commutative algebra, combinatorics and optimization. We study the
normality and the Gorenstein property-as well as the canonical module and the
a-invariant-of Rees algebras and subrings arising from linear optimization
problems. In particular, we study algebraic properties of edge ideals and
algebras associated to uniform clutters with the max-flow min-cut property or
the packing property. We also study algebraic properties of symbolic Rees
algebras of edge ideals of graphs, edge ideals of clique clutters of
comparability graphs, and Stanley-Reisner rings.Comment: PhD thesis, Cinvestav-IPN, June 201
Clique-transversal sets and weak 2-colorings in graphs of small maximum degree
Graphs and Algorithm
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