6,999 research outputs found
The complexity of dominating set reconfiguration
Suppose that we are given two dominating sets and of a graph
whose cardinalities are at most a given threshold . Then, we are asked
whether there exists a sequence of dominating sets of between and
such that each dominating set in the sequence is of cardinality at most
and can be obtained from the previous one by either adding or deleting
exactly one vertex. This problem is known to be PSPACE-complete in general. In
this paper, we study the complexity of this decision problem from the viewpoint
of graph classes. We first prove that the problem remains PSPACE-complete even
for planar graphs, bounded bandwidth graphs, split graphs, and bipartite
graphs. We then give a general scheme to construct linear-time algorithms and
show that the problem can be solved in linear time for cographs, trees, and
interval graphs. Furthermore, for these tractable cases, we can obtain a
desired sequence such that the number of additions and deletions is bounded by
, where is the number of vertices in the input graph
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
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Isomorphism of graph classes related to the circular-ones property
We give a linear-time algorithm that checks for isomorphism between two 0-1
matrices that obey the circular-ones property. This algorithm leads to
linear-time isomorphism algorithms for related graph classes, including Helly
circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and
convex-round graphs.Comment: 25 pages, 9 figure
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