556 research outputs found
Bounds on several versions of restrained domination number
We investigate several versions of restraineddomination numbers and present new bounds on these parameters. We generalize theconcept of restrained domination and improve some well-known bounds in the literature.In particular, for a graph of order and minimum degree , we prove thatthe restrained double domination number of is at most . In addition,for a connected cubic graph of order we show thatthe total restrained domination number of is at least andthe restrained double domination number of is at least
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 , ⌋
Complexity and approximation ratio of semitotal domination in graphs
A set is a semitotal dominating set of a graph if
it is a dominating set of and
every vertex in is within distance 2 of another vertex of . The
semitotal domination number is the minimum
cardinality of a semitotal dominating set of .
We show that the semitotal domination problem is
APX-complete for bounded-degree graphs, and the semitotal domination problem in any graph of maximum degree can be approximated with an approximation
ratio of
Vertex Sequences in Graphs
We consider a variety of types of vertex sequences, which are defined in terms of a requirement that the next vertex in the sequence must meet. For example, let S = (v1, v2, …, vk ) be a sequence of distinct vertices in a graph G such that every vertex vi in S dominates at least one vertex in V that is not dominated by any of the vertices preceding it in the sequence S. Such a sequence of maximal length is called a dominating sequence since the set {v1, v2, …, vk } must be a dominating set of G. In this paper we survey the literature on dominating and other related sequences, and propose for future study several new types of vertex sequences, which suggest the beginning of a theory of vertex sequences in graphs
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