Unanimous and Strategy-Proof Probabilistic Rules for Single-Peaked Preference Profiles on Graphs

Finitely many agents have preferences on a finite set of alternatives, single-peaked with respect to a connected graph with these alternatives as vertices. A probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles at which all peaks are on leaves of the tree and, thus, extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leaves. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation.

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Posterior Probability on Finite Set

Formalized Mathematics ◽

10.2478/v10037-012-0030-0 ◽

2012 ◽

Vol 20 (4) ◽

pp. 257-263

Author(s):

Hiroyuki Okazaki

Keyword(s):

Probability Distribution ◽

Probability Measure ◽

Posterior Probability ◽

Probability Distributions ◽

Sample Space ◽

Finite Sample ◽

Finite Set

Summary In [14] we formalized probability and probability distribution on a finite sample space. In this article first we propose a formalization of the class of finite sample spaces whose element’s probability distributions are equivalent with each other. Next, we formalize the probability measure of the class of sample spaces we have formalized above. Finally, we formalize the sampling and posterior probability.

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The Structure of Locally Finite Two-Connected Graphs

The Electronic Journal of Combinatorics ◽

10.37236/1211 ◽

1995 ◽

Vol 2 (1) ◽

Cited By ~ 12

Author(s):

Carl Droms ◽

Brigitte Servatius ◽

Herman Servatius

Keyword(s):

Connected Graph ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Locally Finite ◽

Connected Graphs ◽

Finite Graphs ◽

Locally Finite Graphs ◽

Necessary And Sufficient ◽

Block Tree ◽

Labeled Tree

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

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First Occurrence in Pairs of Long Words: A Penney-ante Conjecture of Pevzner

Combinatorics Probability Computing ◽

10.1017/s0963548300001656 ◽

1995 ◽

Vol 4 (3) ◽

pp. 279-285

Author(s):

Dudley Stark

Keyword(s):

Probability Distribution ◽

Unique Element ◽

Secondary Maximum ◽

Maximum Value ◽

Random Elements ◽

Finite Set ◽

First Occurrence ◽

Independent And Identically Distributed

Suppose X1, X2,… is a sequence of independent and identically distributed random elements whose values are taken in a finite set S of size |S| ≥ 2 with probability distribution ℙ(X = s) = p(s) > 0 for s ∈ S. Pevzner has conjectured that for every probability distribution ℙ there exists an N > 0 such that for every word A with letters in S whose length is at least N, there exists a second word B of the same length as A, such that the event that B appears before A in the sequence X1, X2,… has greater probability than that of A appearing before B. In this paper it is shown that a distribution ℙ satisfies Pevzner's conclusion if and only if the maximum value of ℙ, p, and the secondary maximum c satisfy the inequality . For |S| = 2 or |S| = 3, the inequality is true and the conjecture holds. If , then the conjecture is true when A is not allowed to consist of pure repetitions of that unique element for which the distribution takes on its mode.

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Some Results on the Eccentric Sequence of Graphs

International Journal of Recent Technology and Engineering - 2 ◽

10.35940/ijrte.d1023.1284s519 ◽

2020 ◽

Vol 8 (4S5) ◽

pp. 55-57

Keyword(s):

Connected Graph ◽

Line Graph ◽

Degree Sequence ◽

Integral Graph

The distance d(u, v) from a vertex u of graph G to a vertex v is the length of a shortest u to v path. The degree of the vertex u is the number of vertices at distance one. The sequence of numbers of vertices having 0,1,2,3,... is called the degree sequence, which is the list of degrees of vertices of G arranged in non-decreasing order. The eccentricity e(a) of a is the distance of a farthest vertex from a. Let G be a connected graph. The Eccentric Sequence of G is the list of the eccentricities of its vertices arranged in non-decreasing order. In this paper, we characterize the eccentric sequence of some of the derived graphs namely the line graph of the integral graph , the eccentric sequence of Mycieleskian of a graph.

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Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph

Complexity ◽

10.1155/2020/6267072 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Jia-Bao Liu ◽

Ali Zafari

Keyword(s):

Connected Graph ◽

Line Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Strong Metric Dimension ◽

Vertex Set

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

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Forbidden Triples Generating a Finite set of 3-Connected Graphs

The Electronic Journal of Combinatorics ◽

10.37236/3255 ◽

2015 ◽

Vol 22 (3) ◽

Cited By ~ 1

Author(s):

Yoshimi Egawa ◽

Jun Fujisawa ◽

Michitaka Furuya ◽

Michael D Plummer ◽

Akira Saito

Keyword(s):

Connected Graph ◽

Induced Subgraph ◽

Connected Graphs ◽

Finite Set

For a graph $G$ and a set $\mathcal{F}$ of connected graphs, $G$ is said be $\mathcal{F}$-free if $G$ does not contain any member of $\mathcal{F}$ as an induced subgraph. We let $\mathcal{G} _{3}(\mathcal{F})$ denote the set of all $3$-connected $\mathcal{F}$-free graphs. This paper is concerned with sets $\mathcal{F}$ of connected graphs such that $|\mathcal{F}|=3$ and $\mathcal{G} _{3}(\mathcal{F})$ is finite. Among other results, we show that for an integer $m\geq 3$ and a connected graph $T$ of order greater than or equal to $4$, $\mathcal{G} _{3}(\{K_{4},K_{2,m},T\})$ is finite if and only if $T$ is a path of order $4$ or $5$.

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Quantization for a Mixture of Uniform Distributions Associated with Probability Vectors

Uniform distribution theory ◽

10.2478/udt-2020-0006 ◽

2020 ◽

Vol 15 (1) ◽

pp. 105-142

Author(s):

Mrinal Kanti Roychowdhury ◽

Wasiela Salinas

Keyword(s):

Probability Distribution ◽

Probability Distributions ◽

Discrete Distribution ◽

Uniform Distributions ◽

Basic Goal ◽

Optimal Quantization ◽

Finite Set ◽

Mixed Distributions

AbstractThe basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this paper, we investigate the optimal quantization for three different mixed distributions generated by uniform distributions associated with probability vectors.

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Connectivity

Graph Theory for Operations Research and Management ◽

10.4018/978-1-4666-2661-4.ch004 ◽

2013 ◽

pp. 37-47

Author(s):

Mahtab Hosseininia ◽

Faraz Dadgostari

Keyword(s):

Connected Graph ◽

Directed Graphs ◽

Graph Connectivity ◽

Breadth First Search ◽

Undirected Graphs ◽

Vertex Connectivity ◽

Edge Connectivity ◽

Graph Traversal ◽

Path Component ◽

Block Tree

In this chapter, the concept of graph connectivity is introduced. In the first section, some concepts such as walk, path, component and connected graph are defined, and connectedness of a graph from the viewpoint of vertex connectivity, and also, edge connectivity are discussed. Then, blocks and block tree of graphs are illustrated. In addition, connectivity in directed graphs is introduced. Furthermore, in the last section, two graph traversal algorithms, depth first search and breadth first search, are described to investigate the connectedness of directed and undirected graphs.

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