3 research outputs found
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
The Parameterized Complexity of Centrality Improvement in Networks
The centrality of a vertex v in a network intuitively captures how important
v is for communication in the network. The task of improving the centrality of
a vertex has many applications, as a higher centrality often implies a larger
impact on the network or less transportation or administration cost. In this
work we study the parameterized complexity of the NP-complete problems
Closeness Improvement and Betweenness Improvement in which we ask to improve a
given vertex' closeness or betweenness centrality by a given amount through
adding a given number of edges to the network. Herein, the closeness of a
vertex v sums the multiplicative inverses of distances of other vertices to v
and the betweenness sums for each pair of vertices the fraction of shortest
paths going through v. Unfortunately, for the natural parameter "number of
edges to add" we obtain hardness results, even in rather restricted cases. On
the positive side, we also give an island of tractability for the parameter
measuring the vertex deletion distance to cluster graphs
Exact exponential algorithms to find a tropical connected set of minimum size
The input of the Tropical Connected Set problem is a vertexcolored graph G = (V, E) and the task is to find a connected subset S ⊆ V of minimum size such that each color of G appears in S. This problem is known to be NP-complete, even when restricted to trees of height at most three. We show that Tropical Connected Set on trees has no subexponential-time algorithm unless the Exponential Time Hypothesis fails. This motivates the study of exact exponential algorithms to solve Tropical Connected Set. We present an O∗(1.5359n) time algorithm for general graphs and an O∗(1.2721n) time algorithm for trees.SCOPUS: cp.kinfo:eu-repo/semantics/publishe