Abstract
This article presents two methods to sample uniform subtrees from graphs using Metropolis-Hastings algorithms. One method is an independent Metropolis-Hastings and the other one is a type of add-and-delete MCMC.
 In addition to the theoretical contributions, we present simulation studies which confirm the theoretical convergence results on our methods by monitoring the convergence of our Markov chains to the equilibrium distribution.
Highlights
This article presents two methods to sample uniform subtrees from graphs using Metropolis-Hastings algorithms
A graph without any cycle is a forest; a tree T (VT, MT ) is a connected forest where the order of a tree is its number of vertices |VT | and the tree size is its number of edges |MT |
We were careful to generate grids with different structures for the same graph, i.e any vertex in the first grid can be connected to another vertex only in a horizontal or vertical direction so the maximum number of edges touching a vertex is 4 whereas any vertex in the second grid can touch a maximum number of 8 neighbour vertices since it is possible for it to connect diagonally, this is justified because we want to detect the behaviour of the sampling methods with different structures
Summary
A graph is a heavily used data structure in the world of algorithms and there are numerous applications of it in computer science like networks of communication, data organization, computational devices and the flow of computation. Let G(V, M ) be a graph consists of V as its finite set of vertices and M is a finite set of ordered pairs of the form (u, v) called edges. A graph is called regular if each vertex has the same number of neighbours; i.e. every vertex has the same degree. A special case of regular graphs is called complete graphs where any vertex in the graph is connected with all other vertices. A graph is connected if for every pair (x, y) of distinct vertices there is a path from x to y. A graph without any cycle is a forest; a tree T (VT , MT ) is a connected forest where the order of a tree is its number of vertices |VT | and the tree size is its number of edges |MT |.
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