4 research outputs found
Algorithmic Aspects of Secure Connected Domination in Graphs
Let be a simple, undirected and connected graph. A connected
dominating set is a secure connected dominating set of , if
for each , there exists such that
and the set is a connected dominating set
of . The minimum size of a secure connected dominating set of denoted by
, is called the secure connected domination number of .
Given a graph and a positive integer the Secure Connected Domination
(SCDM) problem is to check whether has a secure connected dominating set
of size at most In this paper, we prove that the SCDM problem is
NP-complete for doubly chordal graphs, a subclass of chordal graphs. We
investigate the complexity of this problem for some subclasses of bipartite
graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite
and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem
is to find a secure connected dominating set of minimum size in the input
graph. We propose a - approximation algorithm for MSCDS,
where is the maximum degree of the input graph and prove
that MSCDS cannot be approximated within for any unless even for
bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs
with
On Computing the Average Distance for Some Chordal-Like Graphs
The Wiener index of a graph G is the sum of all its distances. Up to renormalization, it is also the average distance in G. The problem of computing this parameter has different applications in chemistry and networks. We here study when it can be done in truly subquadratic time (in the size n+m of the input) on n-vertex m-edge graphs. Our main result is a complete answer to this question, assuming the Strong Exponential-Time Hypothesis (SETH), for all the hereditary subclasses of chordal graphs. Interestingly, the exact same result also holds for the diameter problem. The case of non-hereditary chordal subclasses happens to be more challenging. For the chordal Helly graphs we propose an intricate O?(m^{3/2})-time algorithm for computing the Wiener index, where m denotes the number of edges. We complete our results with the first known linear-time algorithm for this problem on the dually chordal graphs. The former algorithm also computes the median set
Graph Algorithms and Applications
The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity