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In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It ...
In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. [1] A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.
Two different tree-decompositions of the same graph. The width of a tree decomposition is the size of its largest set X i minus one. The treewidth tw(G) of a graph G is the minimum width among all possible tree decompositions of G. In this definition, the size of the largest set is diminished by one in order to make the treewidth of a tree ...
The subset of edges that have one endpoint in each side is called a cut-set. When a cut-set forms a complete bipartite graph, its cut is called a split. Thus, a split can be described as a partition of the vertices of the graph into two subsets X and Y, such that every neighbor of X in Y is adjacent to every neighbor of Y in X. [2]
Some examples of basic tools in Illustrator are selection tools, paintbrush tools, pen tools, pencil tools e.g. Selection tools are used to the layout, adjust, and organize the artwork by selecting, positioning, and stacking objects accurately. Moreover, selection tools can group, lock or hide, and measure objects. [24]
A cutpoint, cut vertex, or articulation point of a graph G is a vertex that is shared by two or more blocks. The structure of the blocks and cutpoints of a connected graph can be described by a tree called the block-cut tree or BC-tree. This tree has a vertex for each block and for each articulation point of the given graph.
For instance, optimizing the cut size for free group sizes puts all vertices in the same community. Additionally, cut size may be the wrong thing to minimize since a good division is not just one with small number of edges between communities. This motivated the use of Modularity (Q) [13] as a metric to optimize a balanced graph partition.
A graph of groups A is called trivial if A = T is already a tree and there is some vertex v of A such that A v = π 1 (A, A). This is equivalent to the condition that A is a tree and that for every edge e = [u, z] of A (with o(e) = u, t(e) = z) such that u is closer to v than z we have [A z : ω e (A e)] = 1, that is A z = ω e (A e).