scholarly journals The Polytope of Degree Partitions

10.37236/1072 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Amitava Bhattacharya ◽  
S. Sivasubramanian ◽  
Murali K. Srinivasan
Keyword(s):  
Convex Hull ◽  
Linear Inequality ◽  
Degree Sequence ◽  
Extreme Points ◽  
Main Tool ◽  
Simple Graph ◽  
Degree Sequences ◽  
Simple Graphs ◽  
Vertex Set ◽  
Partition Polytope

The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. The polytope of degree partitions (respectively, degree sequences) is the convex hull of degree partitions (respectively, degree sequences) of all simple graphs on the vertex set $[n]$. The polytope of degree sequences has been very well studied. In this paper we study the polytope of degree partitions. We show that adding the inequalities $x_1\geq x_2 \geq \cdots \geq x_n$ to a linear inequality description of the degree sequence polytope yields a linear inequality description of the degree partition polytope and we show that the extreme points of the degree partition polytope are the $2^{n-1}$ threshold partitions (these are precisely those extreme points of the degree sequence polytope that have weakly decreasing coordinates). We also show that the degree partition polytope has $2^{n-2}(2n-3)$ edges and $(n^2 -3n + 12)/2$ facets, for $n\geq 4$. Our main tool is an averaging transformation on real sequences defined by repeatedly averaging over the ascending runs.

PARTIAL RESULT ON HADWIGER'S CONJECTURE

2010 ◽  
Vol 02 (03) ◽  
pp. 413-423 ◽  
Author(s):  
ZI-XIA SONG
Keyword(s):  
Degree Sequence ◽  
Simple Graph ◽  
Partial Result ◽  
Degree Sequences ◽  
Induced Subgraph ◽  
Graphic Sequence ◽  
Hadwiger's Conjecture ◽  
Hadwiger’S Conjecture

Let D = (d1, d2, …, dn) be a graphic sequence with 0 ≤ d1 ≤ d2 ≤ ⋯ ≤ dn. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. We say that D is H-free if no graph in R[D] contains H as an induced subgraph. In this paper, we prove that Hadwiger's Conjecture is true for graphs whose degree sequences are claw-free or [Formula: see text]-free.

scholarly journals On the Swap-Distances of Different Realizations of a Graphical Degree Sequence

2013 ◽  
Vol 22 (3) ◽  
pp. 366-383 ◽  
Author(s):  
PÉTER L. ERDŐS ◽  
ZOLTÁN KIRÁLY ◽  
ISTVÁN MIKLÓS
Keyword(s):  
Social Networks ◽  
Markov Chain ◽  
Degree Sequence ◽  
Upper Bounds ◽  
Simple Graph ◽  
Related Question ◽  
Degree Sequences ◽  
Uniform Sampling ◽  
Integer Sequence ◽  
The Greedy Algorithm

One of the first graph-theoretical problems to be given serious attention (in the 1950s) was the decision whether a given integer sequence is equal to the degree sequence of a simple graph (orgraphical, for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on theswapoperation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another of the same degree sequence. This latter problem has received particular attention in the context of rapidly mixing Markov chain approaches to uniform sampling of all possible realizations of a given degree sequence. (This becomes a matter of interest in the context of the study of large social networks, for example.) Previously there were only crude upper bounds on the shortest possible length of such swap sequences between two realizations. In this paper we develop formulae (Gallai-type identities) for theswap-distances of any two realizations of simple undirected or directed degree sequences. These identities considerably improve the known upper bounds on the swap-distances.

scholarly journals New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

2017 ◽  
Vol 27 (2) ◽  
pp. 186-207
Author(s):  
PÉTER L. ERDŐS ◽  
ISTVÁN MIKLÓS ◽  
ZOLTÁN TOROCZKAI
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Directed Graphs ◽  
Degree Sequence ◽  
Degree Sequences ◽  
Network Modelling ◽  
Split Graph ◽  
Simple Graphs ◽  
Large Classes ◽  
Special Degree

In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.

scholarly journals Galois connections between sets of paths and closure operators in simple graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 1573-1581 ◽  
Author(s):  
Josef Šlapal
Keyword(s):  
Positive Integer ◽  
Digital Images ◽  
Galois Connection ◽  
Simple Graph ◽  
Closure Operators ◽  
Simple Graphs ◽  
Galois Connections ◽  
Digital Plane ◽  
Vertex Set

AbstractFor every positive integer n,we introduce and discuss an isotone Galois connection between the sets of paths of lengths n in a simple graph and the closure operators on the (vertex set of the) graph. We consider certain sets of paths in a particular graph on the digital line Z and study the closure operators associated, in the Galois connection discussed, with these sets of paths. We also focus on the closure operators on the digital plane Z2 associated with a special product of the sets of paths considered and show that these closure operators may be used as background structures on the plane for the study of digital images.

scholarly journals Vertex decomposable graph

2016 ◽  
Vol 99 (113) ◽  
pp. 203-209 ◽  
Author(s):  
N. Hajisharifi ◽  
S. Yassemi
Keyword(s):  
Simple Graph ◽  
Simple Graphs ◽  
Vertex Set ◽  
Vertex Decomposable ◽  
Vertex Sets

Let G be a simple graph on the vertex set V (G) and S = {x11,...,xn1} a subset of V (G). Let m1,...,mn ? 2 be integers and G1,...,Gn connected simple graphs on the vertex sets V (Gi) = {xi1,..., ximi} for i = 1,..., n. The graph G(G1,...,Gn) is obtained from G by attaching Gi to G at the vertex xi1 for i = 1,...,n. We give a characterization of G(G1,...,Gn) for being vertex decomposable. This generalizes a result due to Mousivand, Seyed Fakhari, and Yassemi.

scholarly journals On Fractional Realizations of Graph Degree Sequences

10.37236/3683 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Michael D. Barrus
Keyword(s):  
Degree Sequence ◽  
Convex Polytope ◽  
Simple Graph ◽  
Degree Sequences ◽  
Graph Structure ◽  
Forbidden Subgraphs ◽  
Graph Realization ◽  
Split Graphs

We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope; we characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of threshold and pseudo-split graphs, and we characterize their realizations both in terms of forbidden subgraphs and graph structure.

scholarly journals A Simple Havel–Hakimi Type Algorithm to Realize Graphical Degree Sequences of Directed Graphs

10.37236/338 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Péter L. Erdős ◽  
István Miklós ◽  
Zoltán Toroczkai
Keyword(s):  
Greedy Algorithm ◽  
Directed Graphs ◽  
Degree Sequence ◽  
Finite Sequence ◽  
Simple Graph ◽  
Degree Sequences ◽  
Mcmc Algorithm ◽  
Type Algorithm ◽  
Positive Integers ◽  
The Greedy Algorithm

One of the simplest ways to decide whether a given finite sequence of positive integers can arise as the degree sequence of a simple graph is the greedy algorithm of Havel and Hakimi. This note extends their approach to directed graphs. It also studies cases of some simple forbidden edge-sets. Finally, it proves a result which is useful to design an MCMC algorithm to find random realizations of prescribed directed degree sequences.

scholarly journals Spectra of some Operations on Graphs

2016 ◽  
Vol 11 (9) ◽  
pp. 5654-5660
Author(s):  
Essam EI Seidy ◽  
Salah ElDin Hussein ◽  
Atef Abo Elkher
Keyword(s):  
Direct Product ◽  
Cartesian Product ◽  
Simple Graph ◽  
Simple Graphs ◽  
Cartesian Product Of Graphs ◽  
Vertex Set ◽  
Join Of Graphs ◽  
Product Of Graphs

In this paper, we consider a finite undirected and connected simple graph G(E, V) with vertex set V(G) and edge set E(G).We introduced a new computes the spectra of some operations on simple graphs [union of disjoint graphs, join of graphs, cartesian product of graphs, strong cartesian product of graphs, direct product of graphs].

The Min-Max Load Criteria as a Measure of Machining Fixture Performance

1994 ◽  
Vol 116 (4) ◽  
pp. 500-507 ◽  
Author(s):  
E. C. DeMeter
Keyword(s):  
Convex Hull ◽  
Linear Model ◽  
Model Validation ◽  
Contact Region ◽  
Extreme Points ◽  
Simulation Experiments ◽  
Machining Operations ◽  
The Impact ◽  
External Loads ◽  
The Continuum

Spherical-tipped locators and clamps are often used for the restraint of castings during machining. For structurally rigid castings, contact region deformation and micro-slippage are the predominant modes of workpiece displacement. In turn contact region deformation and micro-slippage are heavily influenced by contact region loading. This paper presents a linear model for predicting the impact of locator and clamp placement on workpiece displacement throughout a series of machining operations. It illustrates how the continuum of external loads exerted on a workpiece during machining can be bounded within a convex hull, and how the extreme points of this hull are used within the model. Finally it describes the simulation experiments which were used for model validation.

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

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