10,999 research outputs found
Some new results on the total domination polynomial of a graph
Let be a simple graph of order . The total dominating set of
is a subset of that every vertex of is adjacent to some
vertices of . The total domination number of is equal to minimum
cardinality of total dominating set in and is denoted by . The
total domination polynomial of is the polynomial
, where is the number
of total dominating sets of of size . A root of is called a
total domination root of . An irrelevant edge of is an edge , such that . In this paper, we
characterize edges possessing this property. Also we obtain some results for
the number of total dominating sets of a regular graph. Finally, we study
graphs with exactly two total domination roots , and
.Comment: 12 pages, 7 figure
On the Domination Polynomials of Friendship Graphs
Let be a simple graph of order . The {\em domination polynomial} of
is the polynomial , where
is the number of dominating sets of of size .
Let be any positive integer and be the Friendship graph with vertices and edges, formed by the join of with . We
study the domination polynomials of this family of graphs, and in particular
examine the domination roots of the family, and find the limiting curve for the
roots. We also show that for every , is not
-unique, that is, there is another non-isomorphic graph with the
same domination polynomial. Also we construct some families of graphs whose
real domination roots are only and . Finally, we conclude by discussing
the domination polynomials of a related family of graphs, the -book graphs
, formed by joining copies of the cycle graph with a common
edge.Comment: 16 pages, 7 figures. New version of paper entitled "On
-equivalence class of friendship graphs
Some families of graphs with no nonzero real domination roots
Let G be a simple graph of order n. The domination polynomial is the
generating polynomial for the number of dominating sets of G of each
cardinality. A root of this polynomial is called a domination root of G.
Obviously 0 is a domination root of every graph G. In the study of the
domination roots of graphs, this naturally raises the question: which graphs
have no nonzero real domination roots? In this paper we present some families
of graphs whose have this property.Comment: 13 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1401.209
On the independent domination polynomial of a graph
An independent dominating set of the simple graph is a vertex
subset that is both dominating and independent in . The independent
domination polynomial of a graph is the polynomial , summed over all independent dominating subsets . A root
of is called an independence domination root. We investigate the
independent domination polynomials of some generalized compound graphs. As
consequences, we construct graphs whose independence domination roots are real.
Also, we consider some certain graphs and study the number of their independent
dominating sets.Comment: 16 pages, 4 figure. arXiv admin note: text overlap with
arXiv:1309.7673 by other author
On the roots of total domination polynomial of graphs
Let be a simple graph of order . The total dominating set of
is a subset of that every vertex of is adjacent to some
vertices of . The total domination number of is equal to minimum
cardinality of total dominating set in and denoted by . The
total domination polynomial of is the polynomial
, where is the number of
total dominating sets of of size . In this paper, we study roots of
total domination polynomial of some graphs. We show that all roots of lie in the circle with center and the radius
, where is the minimum degree of . As a
consequence we prove that if , then every integer root
of lies in the set .Comment: 11 pages, 6 figure
The number of dominating -sets of paths, cycles and wheels
We give a shorter proof of the recurrence relation for the domination
polynomial and for the number of
dominating -sets of the path with vertices. For every positive integers
and numbers are determined solving a problem
posed by S. Alikhani in CID 2015. Moreover, the numbers of dominating -sets
of cycles and of wheels with
vertices are computed.Comment: 13 page
Fantom: A scalable framework for asynchronous distributed systems
We describe \emph{Fantom}, a framework for asynchronous distributed systems.
\emph{Fantom} is based on the Lachesis Protocol~\cite{lachesis01}, which uses
asynchronous event transmission for practical Byzantine fault tolerance (pBFT)
to create a leaderless, scalable, asynchronous Directed Acyclic Graph (DAG).
We further optimize the \emph{Lachesis Protocol} by introducing a
permission-less network for dynamic participation. Root selection cost is
further optimized by the introduction of an n-row flag table, as well as
optimizing path selection by introducing domination relationships.
We propose an alternative framework for distributed ledgers, based on
asynchronous partially ordered sets with logical time ordering instead of
blockchains.
This paper builds upon the original proposed family of \emph{Lachesis-class}
consensus protocols. We formalize our proofs into a model that can be applied
to abstract asynchronous distributed system.Comment: arXiv admin note: substantial text overlap with arXiv:1810.0218
Weighted domination number of cactus graphs
In the paper, we write a linear algorithm for calculating the weighted
domination number of a vertex-weighted cactus. The algorithm is based on the
well known depth first search (DFS) structure. Our algorithm needs less than
additions and -operations where is the number of
vertices and is the number of blocks in the cactus.Comment: 17 pages, figures, submitted to Discussiones Mathematicae Graph
Theor
Total domination polynomials of graphs
Given a graph , a total dominating set is a vertex set that every
vertex of is adjacent to some vertices of and let be the
number of all total dominating sets with size . The total domination
polynomial, defined as ,
recently has been one of the considerable extended research in the field of
domination theory. In this paper, we obtain the vertex-reduction and
edge-reduction formulas of total domination polynomials. As consequences, we
give the total domination polynomials for paths and cycles. Additionally, we
determine the sharp upper bounds of total domination polynomials for trees and
characterize the corresponding graphs attaining such bounds. Finally, we use
the reduction-formulas to investigate the relations between vertex sets and
total domination polynomials in
Recurrence relations and splitting formulas for the domination polynomial
The domination polynomial D(G,x) of a graph G is the generating function of
its dominating sets. We prove that D(G,x) satisfies a wide range of reduction
formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs
and for various special cases. We give splitting formulas for D(G,x) based on
articulation vertices, and more generally, on splitting sets of vertices
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