86,201 research outputs found
All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs
We describe algorithms, based on Avis and Fukuda's reverse search paradigm,
for listing all maximal independent sets in a sparse graph in polynomial time
and delay per output. For bounded degree graphs, our algorithms take constant
time per set generated; for minor-closed graph families, the time is O(n) per
set, and for more general sparse graph families we achieve subquadratic time
per set. We also describe new data structures for maintaining a dynamic vertex
set S in a sparse or minor-closed graph family, and querying the number of
vertices not dominated by S; for minor-closed graph families the time per
update is constant, while it is sublinear for any sparse graph family. We can
also maintain a dynamic vertex set in an arbitrary m-edge graph and test the
independence of the maintained set in time O(sqrt m) per update. We use the
domination data structures as part of our enumeration algorithms.Comment: 10 page
Weighted Well-Covered Claw-Free Graphs
A graph G is well-covered if all its maximal independent sets are of the same
cardinality. Assume that a weight function w is defined on its vertices. Then G
is w-well-covered if all maximal independent sets are of the same weight. For
every graph G, the set of weight functions w such that G is w-well-covered is a
vector space. Given an input claw-free graph G, we present an O(n^6)algortihm,
whose input is a claw-free graph G, and output is the vector space of weight
functions w, for which G is w-well-covered. A graph G is equimatchable if all
its maximal matchings are of the same cardinality. Assume that a weight
function w is defined on the edges of G. Then G is w-equimatchable if all its
maximal matchings are of the same weight. For every graph G, the set of weight
functions w such that G is w-equimatchable is a vector space. We present an
O(m*n^4 + n^5*log(n)) algorithm which receives an input graph G, and outputs
the vector space of weight functions w such that G is w-equimatchable.Comment: 14 pages, 1 figur
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
On the Enumeration of all Minimal Triangulations
We present an algorithm that enumerates all the minimal triangulations of a
graph in incremental polynomial time. Consequently, we get an algorithm for
enumerating all the proper tree decompositions, in incremental polynomial time,
where "proper" means that the tree decomposition cannot be improved by removing
or splitting a bag
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