77,177 research outputs found
On dynamic monopolies of graphs: the average and strict majority thresholds
Let be a graph and
be an assignment of thresholds to the vertices of . A subset of vertices
is said to be a dynamic monopoly corresponding to if the vertices
of can be partitioned into subsets such that
and for any , each vertex in has at least
neighbors in . Dynamic monopolies are in fact
modeling the irreversible spread of influence in social networks. In this paper
we first obtain a lower bound for the smallest size of any dynamic monopoly in
terms of the average threshold and the order of graph. Also we obtain an upper
bound in terms of the minimum vertex cover of graphs. Then we derive the upper
bound for the smallest size of any dynamic monopoly when the graph
contains at least one odd vertex, where the threshold of any vertex is set
as (i.e. strict majority threshold). This bound
improves the best known bound for strict majority threshold. We show that the
latter bound can be achieved by a polynomial time algorithm. We also show that
is an upper bound for the size of strict majority dynamic
monopoly, where stands for the matching number of . Finally, we
obtain a basic upper bound for the smallest size of any dynamic monopoly, in
terms of the average threshold and vertex degrees. Using this bound we derive
some other upper bounds
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Relating threshold tolerance graphs to other graph classes
A graph G=(V, E) is a threshold tolerance if it is possible to associate weights and tolerances with each node of G so that two nodes are adjacent exactly when the sum of their weights exceeds either one of their tolerances. Threshold tolerance graphs are a special case of the well-known class of tolerance graphs and generalize the class of threshold graphs which are also extensively studied in literature. In this note we relate the threshold tolerance graphs with other important graph classes. In particular we show that threshold tolerance graphs are a proper subclass of co-strongly chordal graphs and strictly include the class of co-interval graphs. To this purpose, we exploit the relation with another graph class, min leaf power graphs (mLPGs)
On Connectivity in Random Graph Models with Limited Dependencies
For any positive edge density , a random graph in the Erd\H{o}s-Renyi
model is connected with non-zero probability, since all edges are
mutually independent. We consider random graph models in which edges that do
not share endpoints are independent while incident edges may be dependent and
ask: what is the minimum probability , such that for any distribution
(in this model) on graphs with vertices in which each
potential edge has a marginal probability of being present at least ,
a graph drawn from is connected with non-zero probability?
As it turns out, the condition ``edges that do not share endpoints are
independent'' needs to be clarified and the answer to the question above is
sensitive to the specification. In fact, we formalize this intuitive
description into a strict hierarchy of five independence conditions, which we
show to have at least three different behaviors for the threshold .
For each condition, we provide upper and lower bounds for . In the
strongest condition, the coloring model (which includes, e.g., random geometric
graphs), we show that for
, proving a conjecture by Badakhshian, Falgas-Ravry, and
Sharifzadeh. This separates the coloring models from the weaker independence
conditions we consider, as there we prove that . In stark
contrast to the coloring model, for our weakest independence condition --
pairwise independence of non-adjacent edges -- we show that lies
within of the threshold for completely arbitrary
distributions.Comment: 35 pages, 6 figures. [v2] adds related work and is intended as a full
version accompanying the version to appear at RANDOM'2
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