61 research outputs found
More applications of the d-neighbor equivalence: acyclicity and connectivity constraints
In this paper, we design a framework to obtain efficient algorithms for
several problems with a global constraint (acyclicity or connectivity) such as
Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree,
Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that
solves all these problems and whose running time is upper bounded by
, , and where is respectively the clique-width,
-rank-width, rank-width and maximum induced matching width of a
given decomposition. Our meta-algorithm simplifies and unifies the known
algorithms for each of the parameters and its running time matches
asymptotically also the running times of the best known algorithms for basic
NP-hard problems such as Vertex Cover and Dominating Set. Our framework is
based on the -neighbor equivalence defined in [Bui-Xuan, Telle and
Vatshelle, TCS 2013]. The results we obtain highlight the importance of this
equivalence relation on the algorithmic applications of width measures.
We also prove that our framework could be useful for -hard problems
parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For
these latter problems, we obtain , and time
algorithms where is respectively the clique-width, the
-rank-width and the rank-width of the input graph
An Upper Bound on the Size of Obstructions for Bounded Linear Rank-Width
We provide a doubly exponential upper bound in on the size of forbidden
pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field
of linear rank-width at most . As a corollary, we obtain a
doubly exponential upper bound in on the size of forbidden vertex-minors
for graphs of linear rank-width at most . This solves an open question
raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear
rank-width at most . European J. Combin., 41:242--257, 2014]. We also give a
doubly exponential upper bound in on the size of forbidden minors for
matroids representable over a fixed finite field of path-width at most .
Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on
the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series
B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded
path-width. To adapt this notion into linear rank-width, it is necessary to
well define partial pieces of graphs and merging operations that fit to
pivot-minors. Using the algebraic operations introduced by Courcelle and
Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we
define boundaried -labelled graphs and prove similar structure theorems for
pivot-minor and linear rank-width.Comment: 28 pages, 1 figur
Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions
In the companion paper [Linear rank-width of distance-hereditary graphs I. A
polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a
characterization of the linear rank-width of distance-hereditary graphs, from
which we derived an algorithm to compute it in polynomial time. In this paper,
we investigate structural properties of distance-hereditary graphs based on
this characterization.
First, we prove that for a fixed tree , every distance-hereditary graph of
sufficiently large linear rank-width contains a vertex-minor isomorphic to .
We extend this property to bigger graph classes, namely, classes of graphs
whose prime induced subgraphs have bounded linear rank-width. Here, prime
graphs are graphs containing no splits. We conjecture that for every tree ,
every graph of sufficiently large linear rank-width contains a vertex-minor
isomorphic to . Our result implies that it is sufficient to prove this
conjecture for prime graphs.
For a class of graphs closed under taking vertex-minors, a graph
is called a vertex-minor obstruction for if but all of
its proper vertex-minors are contained in . Secondly, we provide, for
each , a set of distance-hereditary graphs that contains all
distance-hereditary vertex-minor obstructions for graphs of linear rank-width
at most . Also, we give a simpler way to obtain the known vertex-minor
obstructions for graphs of linear rank-width at most .Comment: 38 pages, 13 figures, 1 table, revised journal version. A preliminary
version of Section 5 appeared in the proceedings of WG1
Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm
Linear rank-width is a linearized variation of rank-width, and it is deeply
related to matroid path-width. In this paper, we show that the linear
rank-width of every -vertex distance-hereditary graph, equivalently a graph
of rank-width at most , can be computed in time , and a linear layout witnessing the linear rank-width can be computed with
the same time complexity. As a corollary, we show that the path-width of every
-element matroid of branch-width at most can be computed in time
, provided that the matroid is given by an
independent set oracle.
To establish this result, we present a characterization of the linear
rank-width of distance-hereditary graphs in terms of their canonical split
decompositions. This characterization is similar to the known characterization
of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex
separation and search number of a graph. Inf. Comput., 113(1):50--79, 1994].
However, different from forests, it is non-trivial to relate substructures of
the canonical split decomposition of a graph with some substructures of the
given graph. We introduce a notion of `limbs' of canonical split
decompositions, which correspond to certain vertex-minors of the original
graph, for the right characterization.Comment: 28 pages, 3 figures, 2 table. A preliminary version appeared in the
proceedings of WG'1
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
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
An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and
Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B,
96(4):514--528, 2006]. Motivated from recent development on graph modification
problems regarding classes of graphs of bounded treewidth or pathwidth, we
study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex
Deletion). In the LRW1-Vertex Deletion problem, given an -vertex graph
and a positive integer , we want to decide whether there is a set of at most
vertices whose removal turns into a graph of linear rankwidth at most
and find such a vertex set if one exists. While the meta-theorem of
Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved
in time for some function , it is not clear whether this
problem allows a running time with a modest exponential function.
We first establish that LRW1-Vertex Deletion can be solved in time . The major obstacle to this end is how to handle a long
induced cycle as an obstruction. To fix this issue, we define necklace graphs
and investigate their structural properties. Later, we reduce the polynomial
factor by refining the trivial branching step based on a cliquewidth expression
of a graph, and obtain an algorithm that runs in time . We also prove that the running time cannot be improved to under the Exponential Time Hypothesis assumption. Lastly,
we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201
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
- …