346 research outputs found
Algorithmic Complexity of Isolate Secure Domination in Graphs
A dominating set is an Isolate Dominating Set (IDS) if the induced
subgraph has at least one isolated vertex. In this paper, we initiate
the study of new domination parameter called, isolate secure domination. An
isolate dominating set is an isolate secure dominating set
(ISDS), if for each vertex , there exists a neighboring
vertex of in such that is an IDS
of . The minimum cardinality of an ISDS of is called as an isolate
secure domination number, and is denoted by . Given a graph and a positive integer the ISDM problem is to check whether has an isolate secure dominating set of size at most We prove that ISDM
is NP-complete even when restricted to bipartite graphs and split graphs. We
also show that ISDM can be solved in linear time for graphs of bounded
tree-width.Comment: arXiv admin note: substantial text overlap with arXiv:2002.00002;
text overlap with arXiv:2001.1125
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
Algorithmic complexity of isolate secure domination in graphs
A dominating set S is an Isolate Dominating Set (IDS) if the induced subgraph G[S] has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set S subset of V is an isolate secure dominating set (ISDS), if for each vertex u is an element of V \ S, there exists a neighboring vertex v of u in S such that (S \ {v}) boolean OR {u} is an IDS of G. The minimum cardinality of an ISDS of G is called as an isolate secure domination number, and is denoted by gamma(0s) (G). We give isolate secure domination number of path and cycle graphs. Given a graph G = (V, E) and a positive integer k, the ISDM problem is to check whether G has an isolate secure dominating set of size at most k. We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.Publisher's Versio
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