19,973 research outputs found
Building Damage-Resilient Dominating Sets in Complex Networks against Random and Targeted Attacks
We study the vulnerability of dominating sets against random and targeted
node removals in complex networks. While small, cost-efficient dominating sets
play a significant role in controllability and observability of these networks,
a fixed and intact network structure is always implicitly assumed. We find that
cost-efficiency of dominating sets optimized for small size alone comes at a
price of being vulnerable to damage; domination in the remaining network can be
severely disrupted, even if a small fraction of dominator nodes are lost. We
develop two new methods for finding flexible dominating sets, allowing either
adjustable overall resilience, or dominating set size, while maximizing the
dominated fraction of the remaining network after the attack. We analyze the
efficiency of each method on synthetic scale-free networks, as well as real
complex networks
A Greedy Partition Lemma for Directed Domination
A directed dominating set in a directed graph is a set of vertices of
such that every vertex has an adjacent vertex
in with directed to . The directed domination number of , denoted
by , is the minimum cardinality of a directed dominating set in .
The directed domination number of a graph , denoted , which is
the maximum directed domination number over all orientations of
. The directed domination number of a complete graph was first studied by
Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this
paper we prove a Greedy Partition Lemma for directed domination in oriented
graphs. Applying this lemma, we obtain bounds on the directed domination
number. In particular, if denotes the independence number of a graph
, we show that .Comment: 12 page
The Algorithmic Complexity of Bondage and Reinforcement Problems in bipartite graphs
Let be a graph. A subset is a dominating set if
every vertex not in is adjacent to a vertex in . The domination number
of , denoted by , is the smallest cardinality of a dominating set
of . The bondage number of a nonempty graph is the smallest number of
edges whose removal from results in a graph with domination number larger
than . The reinforcement number of is the smallest number of
edges whose addition to results in a graph with smaller domination number
than . In 2012, Hu and Xu proved that the decision problems for the
bondage, the total bondage, the reinforcement and the total reinforcement
numbers are all NP-hard in general graphs. In this paper, we improve these
results to bipartite graphs.Comment: 13 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:1109.1657; and text overlap with arXiv:1204.4010 by other author
Maximum Number of Minimum Dominating and Minimum Total Dominating Sets
Given a connected graph with domination (or total domination) number
\gamma>=2, we ask for the maximum number m_\gamma and m_{\gamma,T} of
dominating and total dominating sets of size \gamma. An exact answer is
provided for \gamma=2and lower bounds are given for \gamma>=3.Comment: 6 page
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