34 research outputs found
On some simplicial elimination schemes for chordal graphs
We present here some results on particular elimination schemes for chordal
graphs, namely we show that for any chordal graph we can construct in linear
time a simplicial elimination scheme starting with a pending maximal clique
attached via a minimal separator maximal (resp. minimal) under inclusion among
all minimal separators
On a class of intersection graphs
Given a directed graph D = (V,A) we define its intersection graph I(D) =
(A,E) to be the graph having A as a node-set and two nodes of I(D) are adjacent
if their corresponding arcs share a common node that is the tail of at least
one of these arcs. We call these graphs facility location graphs since they
arise from the classical uncapacitated facility location problem. In this paper
we show that facility location graphs are hard to recognize and they are easy
to recognize when the graph is triangle-free. We also determine the complexity
of the vertex coloring, the stable set and the facility location problems on
that class
On the Enumeration of Minimal Dominating Sets and Related Notions
A dominating set in a graph is a subset of its vertex set such that each
vertex is either in or has a neighbour in . 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 algorithm for
the Trans-Enum problem if and only if there exists one for the Dom-Enum
problem. Secondly, we study the Dom-Enum problem in some graph classes. We give
an output-polynomial time algorithm for the Dom-Enum problem in split graphs,
and introduce the completion of a graph to obtain an output-polynomial time
algorithm for the Dom-Enum problem in -free chordal graphs, a proper
superclass of split graphs. Finally, we investigate the complexity of the
enumeration of (inclusion-wise) minimal connected dominating sets and minimal
total dominating sets of graphs. We show that there exists an output-polynomial
time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if
and only if there exists one for the following enumeration problems: minimal
total dominating sets, minimal total dominating sets in split graphs, minimal
connected dominating sets in split graphs, minimal dominating sets in
co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio
A Note on Graphs of Linear Rank-Width 1
We prove that a connected graph has linear rank-width 1 if and only if it is
a distance-hereditary graph and its split decomposition tree is a path. An
immediate consequence is that one can decide in linear time whether a graph has
linear rank-width at most 1, and give an obstruction if not. Other immediate
consequences are several characterisations of graphs of linear rank-width 1. In
particular a connected graph has linear rank-width 1 if and only if it is
locally equivalent to a caterpillar if and only if it is a vertex-minor of a
path [O-joung Kwon and Sang-il Oum, Graphs of small rank-width are pivot-minors
of graphs of small tree-width, arxiv:1203.3606] if and only if it does not
contain the co-K_2 graph, the Net graph and the 5-cycle graph as vertex-minors
[Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for
linear rank-width at most 1, arxiv:1106.2533].Comment: 9 pages, 2 figures. Not to be publishe
Algorithmic Aspects of a General Modular Decomposition Theory
A new general decomposition theory inspired from modular graph decomposition
is presented. This helps unifying modular decomposition on different
structures, including (but not restricted to) graphs. Moreover, even in the
case of graphs, the terminology ``module'' not only captures the classical
graph modules but also allows to handle 2-connected components, star-cutsets,
and other vertex subsets. The main result is that most of the nice algorithmic
tools developed for modular decomposition of graphs still apply efficiently on
our generalisation of modules. Besides, when an essential axiom is satisfied,
almost all the important properties can be retrieved. For this case, an
algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is
generalised and yields a very efficient solution to the associated
decomposition problem
A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs
An output-polynomial algorithm for the listing of minimal dominating sets in
graphs is a challenging open problem and is known to be equivalent to the
well-known Transversal problem which asks for an output-polynomial algorithm
for listing the set of minimal hitting sets in hypergraphs. We give a
polynomial delay algorithm to list the set of minimal dominating sets in
chordal graphs, an important and well-studied graph class where such an
algorithm was open for a while.Comment: 13 pages, 1 figure, submitte
A Note On Computing Set Overlap Classes
Let be a finite set of elements and a family of subsets of Two sets and of
overlap if and Two sets
are in the same overlap class if there is a series of
sets of in which each overlaps. In this note, we focus
on efficiently identifying all overlap classes in
time. We thus revisit the clever algorithm of Dahlhaus of which we give a clear
presentation and that we simplify to make it practical and implementable in its
real worst case complexity. An useful variant of Dahlhaus's approach is also
explained
Efficient enumeration of maximal split subgraphs and sub-cographs and related classes
In this paper, we are interested in algorithms that take in input an
arbitrary graph , and that enumerate in output all the (inclusion-wise)
maximal "subgraphs" of which fulfil a given property . All over this
paper, we study several different properties , and the notion of subgraph
under consideration (induced or not) will vary from a result to another.
More precisely, we present efficient algorithms to list all maximal split
subgraphs, sub-cographs and some subclasses of cographs of a given input graph.
All the algorithms presented here run in polynomial delay, and moreover for
split graphs it only requires polynomial space. In order to develop an
algorithm for maximal split (edge-)subgraphs, we establish a bijection between
the maximal split subgraphs and the maximal independent sets of an auxiliary
graph. For cographs and some subclasses , the algorithms rely on a framework
recently introduced by Conte & Uno called Proximity Search. Finally we consider
the extension problem, which consists in deciding if there exists a maximal
induced subgraph satisfying a property that contains a set of prescribed
vertices and that avoids another set of vertices. We show that this problem is
NP-complete for every "interesting" hereditary property . We extend the
hardness result to some specific edge version of the extension problem
Polynomial Delay Algorithm for Minimal Chordal Completions
Motivated by the problem of enumerating all tree decompositions of a graph, we consider in this article the problem of listing all the minimal chordal completions of a graph. In [Carmeli et al., 2020] (Pods 2017) Carmeli et al. proved that all minimal chordal completions or equivalently all proper tree decompositions of a graph can be listed in incremental polynomial time using exponential space. The total running time of their algorithm is quadratic in the number of solutions and the existence of an algorithm whose complexity depends only linearly on the number of solutions remained open. We close this question by providing a polynomial delay algorithm to solve this problem which, moreover, uses polynomial space.
Our algorithm relies on Proximity Search, a framework recently introduced by Conte and Uno [Conte and Uno, 2019] (Stoc 2019) which has been shown powerful to obtain polynomial delay algorithms, but generally requires exponential space. In order to obtain a polynomial space algorithm for our problem, we introduce a new general method called canonical path reconstruction to design polynomial delay and polynomial space algorithms based on proximity search
Polynomial Delay Algorithm for Minimal Chordal Completions
Motivated by the problem of enumerating all tree decompositions of a graph, we consider in this article the problem of listing all the minimal chordal completions of a graph. In [Carmeli et al., 2020] (Pods 2017) Carmeli et al. proved that all minimal chordal completions or equivalently all proper tree decompositions of a graph can be listed in incremental polynomial time using exponential space. The total running time of their algorithm is quadratic in the number of solutions and the existence of an algorithm whose complexity depends only linearly on the number of solutions remained open. We close this question by providing a polynomial delay algorithm to solve this problem which, moreover, uses polynomial space.
Our algorithm relies on Proximity Search, a framework recently introduced by Conte and Uno [Conte and Uno, 2019] (Stoc 2019) which has been shown powerful to obtain polynomial delay algorithms, but generally requires exponential space. In order to obtain a polynomial space algorithm for our problem, we introduce a new general method called canonical path reconstruction to design polynomial delay and polynomial space algorithms based on proximity search