8,037 research outputs found
Total 2-domination of proper interval graphs
A set of vertices W of a graph G is a total k-dominating set when every vertex of G has at least k neighbors in W. In a recent article, Chiarelli et al. (2019) prove that a total k-dominating set can be computed in O(n3k) time when G is a proper interval graph with n vertices and m edges. In this note we reduce the time complexity to O(m) for k=2.Fil: Soulignac, Francisco Juan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación; Argentin
Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness
Domination and coloring are two classic problems in graph theory. The major
focus of this paper is the CD-COLORING problem which combines the flavours of
domination and colouring. Let be an undirected graph. A proper vertex
coloring of is a if each color class has a dominating vertex
in . The minimum integer for which there exists a of
using colors is called the cd-chromatic number, . A set
is a total dominating set if any vertex in has a neighbor
in . The total domination number, of is the minimum
integer such that has a total dominating set of size . A set
is a if no two vertices in lie at a
distance 2 in . The separated-cluster number, , of is the
maximum integer such that has a separated-cluster of size .
In this paper, first we explore the connection between CD-COLORING and TOTAL
DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on
triangle-free -regular graphs for each fixed integer . We also
study the relationship between the parameters and .
Analogous to the well-known notion of `perfectness', here we introduce the
notion of `cd-perfectness'. We prove a sufficient condition for a graph to
be cd-perfect (i.e. , for any induced subgraph
of ) which is also necessary for certain graph classes (like triangle-free
graphs). Here, we propose a generalized framework via which we obtain several
exciting consequences in the algorithmic complexities of special graph classes.
In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is
polynomially solvable for interval graphs
Algorithmic aspects of disjunctive domination in graphs
For a graph , a set is called a \emph{disjunctive
dominating set} of if for every vertex , is either
adjacent to a vertex of or has at least two vertices in at distance
from it. The cardinality of a minimum disjunctive dominating set of is
called the \emph{disjunctive domination number} of graph , and is denoted by
. The \textsc{Minimum Disjunctive Domination Problem} (MDDP)
is to find a disjunctive dominating set of cardinality .
Given a positive integer and a graph , the \textsc{Disjunctive
Domination Decision Problem} (DDDP) is to decide whether has a disjunctive
dominating set of cardinality at most . In this article, we first propose a
linear time algorithm for MDDP in proper interval graphs. Next we tighten the
NP-completeness of DDDP by showing that it remains NP-complete even in chordal
graphs. We also propose a -approximation
algorithm for MDDP in general graphs and prove that MDDP can not be
approximated within for any unless NP
DTIME. Finally, we show that MDDP is
APX-complete for bipartite graphs with maximum degree
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
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