8,806 research outputs found
Perfect Roman Domination and Unique Response Roman Domination
The idea of enumeration algorithms with polynomial delay is to polynomially
bound the running time between any two subsequent solutions output by the
enumeration algorithm. While it is open for more than four decades if all
minimal dominating sets of a graph can be enumerated in output-polynomial time,
it has recently been proven that pointwise-minimal Roman dominating functions
can be enumerated even with polynomial delay. The idea of the enumeration
algorithm was to use polynomial-time solvable extension problems. We use this
as a motivation to prove that also two variants of Roman dominating functions
studied in the literature, named perfect and unique response, can be enumerated
with polynomial delay. This is interesting since Extension Perfect Roman
Domination is W[1]-complete if parameterized by the weight of the given
function and even W[2]-complete if parameterized by the number vertices
assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability
of extension problems and enumerability with polynomial delay tend to go
hand-in-hand. We achieve our enumeration result by constructing a bijection to
Roman dominating functions, where the corresponding extension problem is
polynomimaltime solvable. Furthermore, we show that Unique Response Roman
Domination is solvable in polynomial time on split graphs, while Perfect Roman
Domination is NP-complete on this graph class, which proves that both
variations, albeit coming with a very similar definition, do differ in some
complexity aspects. This way, we also solve an open problem from the
literature
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
On the Roman domination in the lexicographic product of graphs
AbstractA Roman dominating function of a graph G=(V,E) is a function f:V→{0,1,2} such that every vertex with f(v)=0 is adjacent to some vertex with f(v)=2. The Roman domination number of G is the minimum of w(f)=∑v∈Vf(v) over all such functions. Using a new concept of the so-called dominating couple we establish the Roman domination number of the lexicographic product of graphs. We also characterize Roman graphs among the lexicographic product of graphs
Disjoint Dominating Sets with a Perfect Matching
In this paper, we consider dominating sets and such that and
are disjoint and there exists a perfect matching between them. Let
denote the cardinality of smallest such sets in
(provided they exist, otherwise ). This
concept was introduced in [Klostermeyer et al., Theory and Application of
Graphs, 2017] in the context of studying a certain graph protection problem. We
characterize the trees for which equals a certain
graph protection parameter and for which ,
where is the independence number of . We also further study this
parameter in graph products, e.g., by giving bounds for grid graphs, and in
graphs of small independence number
Theoretical Computer Science and Discrete Mathematics
This book includes 15 articles published in the Special Issue "Theoretical Computer Science and Discrete Mathematics" of Symmetry (ISSN 2073-8994). This Special Issue is devoted to original and significant contributions to theoretical computer science and discrete mathematics. The aim was to bring together research papers linking different areas of discrete mathematics and theoretical computer science, as well as applications of discrete mathematics to other areas of science and technology. The Special Issue covers topics in discrete mathematics including (but not limited to) graph theory, cryptography, numerical semigroups, discrete optimization, algorithms, and complexity
Tracking advanced persistent threats in critical infrastructures through opinion dynamics
Advanced persistent threats pose a serious issue for modern industrial environments, due to their targeted and complex attack vectors that are difficult to detect. This is especially severe in critical infrastructures that are accelerating the integration of IT technologies. It is then essential to further develop effective monitoring and response systems that ensure the continuity of business to face the arising set of cyber-security threats. In this paper, we study the practical applicability of a novel technique based on opinion dynamics, that permits to trace the attack throughout all its stages along the network by correlating different anomalies measured over time, thereby taking the persistence of threats and the criticality of resources into consideration. The resulting information is of essential importance to monitor the overall health of the control system and cor- respondingly deploy accurate response procedures. Advanced Persistent Threat Detection Traceability Opinion Dynamics.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
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