19 research outputs found
Decycling a graph by the removal of a matching: new algorithmic and structural aspects in some classes of graphs
A graph is {\em matching-decyclable} if it has a matching such that
is acyclic. Deciding whether is matching-decyclable is an NP-complete
problem even if is 2-connected, planar, and subcubic. In this work we
present results on matching-decyclability in the following classes: Hamiltonian
subcubic graphs, chordal graphs, and distance-hereditary graphs. In Hamiltonian
subcubic graphs we show that deciding matching-decyclability is NP-complete
even if there are exactly two vertices of degree two. For chordal and
distance-hereditary graphs, we present characterizations of
matching-decyclability that lead to -time recognition algorithms
Hitting forbidden induced subgraphs on bounded treewidth graphs
For a fixed graph , the -IS-Deletion problem asks, given a graph ,
for the minimum size of a set such that does
not contain as an induced subgraph. Motivated by previous work about
hitting (topological) minors and subgraphs on bounded treewidth graphs, we are
interested in determining, for a fixed graph , the smallest function
such that -IS-Deletion can be solved in time assuming the Exponential Time Hypothesis (ETH), where and
denote the treewidth and the number of vertices of the input graph,
respectively.
We show that for every graph on
vertices, and that if is a clique or an independent
set. We present a number of lower bounds by generalizing a reduction of Cygan
et al. [MFCS 2014] for the subgraph version. In particular, we show that when
deviates slightly from a clique, the function suffers a sharp
jump: if is obtained from a clique of size by removing one edge, then
. We also show that
when , and this reduction answers an open question of Mi. Pilipczuk
[MFCS 2011] about the function for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant
of the problem, where each vertex of is colored with some color from
and we require to hit only induced copies of with matching colors. In this
case, we determine, under the ETH, the function for every connected
graph on vertices: if the problem can be solved in polynomial
time; if , if is a clique, and otherwise.Comment: 24 pages, 3 figure
On Conflict-Free Cuts: Algorithms and Complexity
One way to define the Matching Cut problem is: Given a graph , is there an
edge-cut of such that is an independent set in the line graph of
? We propose the more general Conflict-Free Cut problem: Together with the
graph , we are given a so-called conflict graph on the edges of
, and we ask for an edge-cutset of that is independent in .
Since conflict-free settings are popular generalizations of classical
optimization problems and Conflict-Free Cut was not considered in the
literature so far, we start the study of the problem. We show that the problem
is -complete even when the maximum degree of is 5 and
is 1-regular. The same reduction implies an exponential lower bound
on the solvability based on the Exponential Time Hypothesis. We also give
parameterized complexity results: We show that the problem is fixed-parameter
tractable with the vertex cover number of as a parameter, and we show
-hardness even when has a feedback vertex set of size one,
and the clique cover number of is the parameter. Since the clique
cover number of is an upper bound on the independence number of
and thus the solution size, this implies -hardness
when parameterized by the cut size. We list polynomial-time solvable cases and
interesting open problems. At last, we draw a connection to a symmetric variant
of SAT.Comment: 13 pages, 3 figure
Exact and Parameterized Algorithms for the Independent Cutset Problem
The Independent Cutset problem asks whether there is a set of vertices in a
given graph that is both independent and a cutset. Such a problem is
-complete even when the input graph is planar and has maximum
degree five. In this paper, we first present a -time
algorithm for the problem. We also show how to compute a minimum independent
cutset (if any) in the same running time. Since the property of having an
independent cutset is MSO-expressible, our main results are concerned with
structural parameterizations for the problem considering parameters that are
not bounded by a function of the clique-width of the input. We present
-time algorithms for the problem considering the following
parameters: the dual of the maximum degree, the dual of the solution size, the
size of a dominating set (where a dominating set is given as an additional
input), the size of an odd cycle transversal, the distance to chordal graphs,
and the distance to -free graphs. We close by introducing the notion of
-domination, which allows us to identify more fixed-parameter tractable
and polynomial-time solvable cases.Comment: 20 pages with references and appendi
Co-Degeneracy and Co-Treewidth: Using the Complement to Solve Dense Instances
Clique-width and treewidth are two of the most important and useful graph parameters, and several problems can be solved efficiently when restricted to graphs of bounded clique-width or treewidth. Bounded treewidth implies bounded clique-width, but not vice versa. Problems like Longest Cycle, Longest Path, MaxCut, Edge Dominating Set, and Graph Coloring are fixed-parameter tractable when parameterized by the treewidth, but they cannot be solved in FPT time when parameterized by the clique-width unless FPT = W[1], as shown by Fomin, Golovach, Lokshtanov, and Saurabh [SIAM J. Comput. 2010, SIAM J. Comput. 2014]. For a given problem that is fixed-parameter tractable when parameterized by treewidth, but intractable when parameterized by clique-width, there may exist infinite families of instances of bounded clique-width and unbounded treewidth where the problem can be solved efficiently. In this work, we initiate a systematic study of the parameters co-treewidth (the treewidth of the complement of the input graph) and co-degeneracy (the degeneracy of the complement of the input graph). We show that Longest Cycle, Longest Path, and Edge Dominating Set are FPT when parameterized by co-degeneracy. On the other hand, Graph Coloring is para-NP-complete when parameterized by co-degeneracy but FPT when parameterized by the co-treewidth. Concerning MaxCut, we give an FPT algorithm parameterized by co-treewidth, while we leave open the complexity of the problem parameterized by co-degeneracy. Additionally, we show that Precoloring Extension is fixed-parameter tractable when parameterized by co-treewidth, while this problem is known to be W[1]-hard when parameterized by treewidth. These results give evidence that co-treewidth is a useful width parameter for handling dense instances of problems for which an FPT algorithm for clique-width is unlikely to exist. Finally, we develop an algorithmic framework for co-degeneracy based on the notion of Bondy-Chvátal closure.publishedVersio