scholarly journals The $t$-Stability Number of a Random Graph

10.37236/331 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Nikolaos Fountoulakis ◽  
Ross J. Kang ◽  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Maximum Degree ◽  
Asymptotic Expression ◽  
Main Tool ◽  
Stable Set ◽  
Stable Sets ◽  
Stability Number ◽  
Expected Number ◽  
Maximum Order ◽  
Edge Probability

Given a graph $G = (V,E)$, a vertex subset $S \subseteq V$ is called $t$-stable (or $t$-dependent) if the subgraph $G[S]$ induced on $S$ has maximum degree at most $t$. The $t$-stability number $\alpha_t(G)$ of $G$ is the maximum order of a $t$-stable set in $G$. The theme of this paper is the typical values that this parameter takes on a random graph on $n$ vertices and edge probability equal to $p$. For any fixed $0 < p < 1$ and fixed non-negative integer $t$, we show that, with probability tending to $1$ as $n\to\infty$, the $t$-stability number takes on at most two values which we identify as functions of $t$, $p$ and $n$. The main tool we use is an asymptotic expression for the expected number of $t$-stable sets of order $k$. We derive this expression by performing a precise count of the number of graphs on $k$ vertices that have maximum degree at most $t$.

scholarly journals On the maximum orders of an induced forest, an induced tree, and a stable set

2014 ◽  
Vol 24 (2) ◽  
pp. 199-215
Author(s):  
Alain Hertz ◽  
Odile Marcotte ◽  
David Schindl
Keyword(s):  
Connected Graph ◽  
Stable Set ◽  
Stability Number ◽  
Maximum Order ◽  
The Stability ◽  
Special Case

Let G be a connected graph, n the order of G, and f (resp. t) the maximum order of an induced forest (resp. tree) in G. We show that f - t is at most n - ?2?n-1?. In the special case where n is of the form a2 + 1 for some even integer a ? 4, f - t is at most n - ?2?n-1?-1. We also prove that these bounds are tight. In addition, letting ? denote the stability number of G, we show that ? - t is at most n + 1- ?2?2n? this bound is also tight.

scholarly journals The t-Improper Chromatic Number of Random Graphs

2009 ◽  
Vol 19 (1) ◽  
pp. 87-98 ◽  
Author(s):  
ROSS J. KANG ◽  
COLIN McDIARMID
Keyword(s):  
Asymptotic Behaviour ◽  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Maximum Degree ◽  
Colour Class ◽  
Edge Probability

We consider the t-improper chromatic number of the Erdős–Rényi random graph Gn,p. The t-improper chromatic number χt(G) is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most t. If t = 0, then this is the usual notion of proper colouring. When the edge probability p is constant, we provide a detailed description of the asymptotic behaviour of χt(Gn,p) over the range of choices for the growth of t = t(n).

scholarly journals A Note on the Majority Dynamics in Inhomogeneous Random Graphs

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Random Graph ◽  
Initial State ◽  
Time Step ◽  
Dynamic Monopoly ◽  
Inhomogeneous Random Graphs ◽  
Assignment Probability ◽  
Edge Probability ◽  
Consensus Reaching Process ◽  
Consensus Reaching ◽  
Inhomogeneous Random Graph

AbstractIn this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph $$K_n$$ K n independently with probability $$p_n(e)$$ p n ( e ) . Each vertex is independently assigned an initial state $$+1$$ + 1 (with probability $$p_+$$ p + ) or $$-1$$ - 1 (with probability $$1-p_+$$ 1 - p + ), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if $$p_+$$ p + is smaller than a threshold, then G will display a unanimous state $$-1$$ - 1 asymptotically almost surely, meaning that the probability of reaching consensus tends to one as $$n\rightarrow \infty $$ n → ∞ . The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph $$p_+$$ p + can be near a half, while in a sparse random graph $$p_+$$ p + has to be vanishing. The size of a dynamic monopoly in G is also discussed.

scholarly journals Coalition formation and history dependence

2020 ◽  
Vol 15 (1) ◽  
pp. 159-197 ◽  
Author(s):  
Bhaskar Dutta ◽  
Hannu Vartiainen
Keyword(s):  
Coalition Formation ◽  
Stable Set ◽  
Transferable Utility ◽  
Stable Sets ◽  
Solution Concepts ◽  
History Dependence ◽  
Von Neumann ◽  
Finite Games ◽  
A Chain ◽  
Indirect Dominance

Farsighted formulations of coalitional formation, for instance, by Harsanyi and Ray and Vohra, have typically been based on the von Neumann–Morgenstern stable set. These farsighted stable sets use a notion of indirect dominance in which an outcome can be dominated by a chain of coalitional “moves” in which each coalition that is involved in the sequence eventually stands to gain. Dutta and Vohra point out that these solution concepts do not require coalitions to make optimal moves. Hence, these solution concepts can yield unreasonable predictions. Dutta and Vohra restricted coalitions to hold common, history‐independent expectations that incorporate optimality regarding the continuation path. This paper extends the Dutta–Vohra analysis by allowing for history‐dependent expectations. The paper provides characterization results for two solution concepts that correspond to two versions of optimality. It demonstrates the power of history dependence by establishing nonemptyness results for all finite games as well as transferable utility partition function games. The paper also provides partial comparisons of the solution concepts to other solutions.

scholarly journals Connectivity for Random Graphs from a Weighted Bridge-Addable Class

10.37236/2596 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Colin McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Planar Graphs ◽  
Additional Assumption ◽  
General Distribution ◽  
Expected Number ◽  
Recent Interest ◽  
Main Conjecture

There has been much recent interest in random graphs sampled uniformly from the $n$-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general bridge-addable class $\cal A$ of graphs -- if a graph is in $\cal A$ and $u$ and $v$ are vertices in different components   then the graph obtained by adding an edge (bridge) between $u$ and $v$ must also be in $\cal A$. Various bounds are known concerning the probability of a random graph from such a   class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added. Here we improve or amplify or generalise these bounds (though we do not resolve the main conjecture). For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider `weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.

The stable set polytope of claw-free graphs with stability number greater than three

2012 ◽  
pp. 47-52
Author(s):  
Anna Galluccio ◽  
Claudio Gentile ◽  
Paolo Ventura
Keyword(s):  
Stable Set ◽  
Stability Number ◽  
Stable Set Polytope

The convex hull of a uniform sample from the interior of a simple d-polytope

1989 ◽  
Vol 26 (02) ◽  
pp. 259-273 ◽  
Author(s):  
Barthold F. Van Wel
Keyword(s):  
Convex Hull ◽  
Asymptotic Expression ◽  
Expected Number ◽  
Simple Polytope ◽  
Uniform Sample ◽  
Sample Points

An asymptotic expression is given for the expected number of vertices of the convex hull of a uniform sample from the interior of a d-dimensional simple polytope. This extends a result derived by Rényi and Sulanke for sample points in the plane.

scholarly journals Maximality in the Farsighted Stable Set

Econometrica ◽  
2019 ◽  
Vol 87 (5) ◽  
pp. 1763-1779 ◽  
Author(s):  
Debraj Ray ◽  
Rajiv Vohra
Keyword(s):  
Stable Set ◽  
Stable Sets ◽  
Von Neumann

Harsanyi (1974) and Ray and Vohra (2015) extended the stable set of von Neumann and Morgenstern to impose farsighted credibility on coalitional deviations. But the resulting farsighted stable set suffers from a conceptual drawback: while coalitional moves improve on existing outcomes, coalitions might do even better by moving elsewhere. Or other coalitions might intervene to impose their favored moves. We show that every farsighted stable set satisfying some reasonable and easily verifiable properties is unaffected by the imposition of these stringent maximality constraints. The properties we describe are satisfied by many, but not all, farsighted stable sets.

scholarly journals An existence theorem for abstract stable sets

1991 ◽  
Vol 51 (3) ◽  
pp. 468-472 ◽  
Author(s):  
Chih Chang ◽  
Gerard J. Chang
Keyword(s):  
Existence Theorem ◽  
Dominance Relation ◽  
Stable Set ◽  
Stable Sets

AbstractWe provide an existence theorem for stable sets which is equivalent to Zorn's lemma and study the connections between the unique stable set for majorization and the stable sets for the dominance relation.

scholarly journals VERY WELL-COVERED GRAPHS OF GIRTH AT LEAST FOUR AND LOCAL MAXIMUM STABLE SET GREEDOIDS

2011 ◽  
Vol 03 (02) ◽  
pp. 245-252 ◽  
Author(s):  
VADIM E. LEVIT ◽  
EUGEN MANDRESCU
Keyword(s):  
Perfect Matching ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
Maximum Cardinality ◽  
Math Program ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Appl Math

A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, and we write S ∈ Ψ(G), if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter Jr. [Vertex packings: structural properties and algorithms, Math. Program.8 (1975) 232–248], proved that any S ∈ Ψ(G) is a subset of a maximum stable set of G. In [Levit and Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discrete Appl. Math.124 (2002) 91–101] we have shown that the family Ψ(T) of a forest T forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while Ψ(G) is a greedoid, were analyzed in [Levit and Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math.132 (2004) 163–174], [Levit and Mandrescu, Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids, Discrete Appl. Math.155 (2007) 2414–2425], [Levit and Mandrescu, Well-covered graphs and greedoids, Proc. 14th Computing: The Australasian Theory Symp. (CATS2008), Wollongong, NSW, Conferences in Research and Practice in Information Technology, Vol. 77 (2008) 89–94], respectively. In this paper we demonstrate that if G is a very well-covered graph of girth ≥4, then the family Ψ(G) is a greedoid if and only if G has a unique perfect matching.

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