2,522 research outputs found
DETERMINATION OF THE RESTRAINED DOMINATION NUMBER ON VERTEX AMALGAMATION AND EDGE AMALGAMATION OF THE PATH GRAPH WITH THE SAME ORDER
Graph theory is a mathematics section that studies discrete objects. One of the concepts studied in graph theory is the restrained dominating set which aims to find the restrained dominating number. This research was conducted by examining the graph operation result of the vertex and edges amalgamation of the path graph in the same order. The method used in this research is the deductive method by using existing theorems to produce new theorems that will be proven deductively true. This research aimed to determine the restrained dominating number in vertex and edges amalgamation of the path graph in the same order. The results obtained from this study are in the form of the theorem about the restrained dominating number of vertex and edges amalgamation of the path graph in the same order, namely: for , ⌋, and for , ⌋
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
Characterizations in Domination Theory
Let G = (V,E) be a graph. A set R is a restrained dominating set (total restrained dominating set, resp.) if every vertex in V − R (V) is adjacent to a vertex in R and (every vertex in V −R) to a vertex in V −R. The restrained domination number of G (total restrained domination number of G), denoted by gamma_r(G) (gamma_tr(G)), is the smallest cardinality of a restrained dominating set (total restrained dominating set) of G. If T is a tree of order n, then gamma_r(T) is greater than or equal to (n+2)/3. We show that gamma_tr(T) is greater than or equal to (n+2)/2. Moreover, we show that if n is congruent to 0 mod 4, then gamma_tr(T) is greater than or equal to (n+2)/2 + 1. We then constructively characterize the extremal trees achieving these lower bounds. Finally, if G is a graph of order n greater than or equal to 2, such that both G and G\u27 are not isomorphic to P_3, then gamma_r(G) + gamma_r(G\u27) is greater than or equal to 4 and less than or equal to n +2. We provide a similar result for total restrained domination and characterize the extremal graphs G of order n achieving these bounds
Empirical Evaluation of Abstract Argumentation: Supporting the Need for Bipolar and Probabilistic Approaches
In dialogical argumentation it is often assumed that the involved parties
always correctly identify the intended statements posited by each other,
realize all of the associated relations, conform to the three acceptability
states (accepted, rejected, undecided), adjust their views when new and correct
information comes in, and that a framework handling only attack relations is
sufficient to represent their opinions. Although it is natural to make these
assumptions as a starting point for further research, removing them or even
acknowledging that such removal should happen is more challenging for some of
these concepts than for others. Probabilistic argumentation is one of the
approaches that can be harnessed for more accurate user modelling. The
epistemic approach allows us to represent how much a given argument is believed
by a given person, offering us the possibility to express more than just three
agreement states. It is equipped with a wide range of postulates, including
those that do not make any restrictions concerning how initial arguments should
be viewed, thus potentially being more adequate for handling beliefs of the
people that have not fully disclosed their opinions in comparison to Dung's
semantics. The constellation approach can be used to represent the views of
different people concerning the structure of the framework we are dealing with,
including cases in which not all relations are acknowledged or when they are
seen differently than intended. Finally, bipolar argumentation frameworks can
be used to express both positive and negative relations between arguments. In
this paper we describe the results of an experiment in which participants
judged dialogues in terms of agreement and structure. We compare our findings
with the aforementioned assumptions as well as with the constellation and
epistemic approaches to probabilistic argumentation and bipolar argumentation
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