# bctree

Block-cut tree graph

## Description

returns the block-cut tree of graph `tree`

= bctree(`G`

)`G`

, such that each node in
`tree`

represents either a biconnected component or cut
vertex of `G`

. A node representing a cut vertex is
connected to all nodes representing biconnected components that contain that cut
vertex.

## Examples

### Compute Block-Cut Tree of Graph

Compute the block-cut tree of a graph, view the resulting node properties, and then highlight the cut vertices in the graph plot.

Create and plot a graph.

s = [1 1 2 2 3 4 4 5 6 6 7 7 8]; t = [2 3 3 4 4 5 7 6 7 10 8 9 9]; G = graph(s,t); p = plot(G);

Compute the block-cut tree of the graph and view the node properties.

tree = bctree(G); tree.Nodes

`ans=`*7×3 table*
IsComponent ComponentIndex CutVertexIndex
___________ ______________ ______________
true 1 0
true 2 0
true 3 0
true 4 0
false 0 4
false 0 6
false 0 7

Plot the block-cut tree using red diamond markers for the nodes that represent cut vertices. The circular nodes represent the biconnected components in the original graph.

p2 = plot(tree,'MarkerSize',9); highlight(p2,5:7,'Marker','d','NodeColor','r')

### Return Node Indices for Block-Cut Tree

Create and plot a graph.

s = [1 1 2 2 3 4 4 5 6 6 7 7 8]; t = [2 3 3 4 4 5 7 6 7 10 8 9 9]; G = graph(s,t); p = plot(G);

Compute the block-cut tree `tr`

of the graph, and specify a second output `ix`

to return the node indices.

[tr,ix] = bctree(G)

tr = graph with properties: Edges: [6x1 table] Nodes: [7x3 table]

`ix = `*1×10*
4 4 4 5 3 6 7 1 1 2

Each index `ix(j)`

indicates the node in the block-cut tree that represents node `j`

in the input graph. For example, node 4 in `tr`

represents a component in `G`

that contains nodes 1, 2, and 3, so the first three entries in `ix`

are all 4.

## Input Arguments

`G`

— Input graph

`graph`

object

Input graph, specified as a `graph`

object. Use `graph`

to create an undirected graph object.

**Example: **`G = graph(1,2)`

## Output Arguments

`tree`

— Block-cut tree graph

`graph`

object

Block-cut tree graph, returned as a `graph`

object.
`tree`

contains a node for each cut vertex in
`G`

and a node for each biconnected component in
`G`

. The nodes table `tree.Nodes`

contains additional node attributes to describe what each node
represents:

`tree.Nodes.IsComponent(i)`

— Value equal to logical`1`

(`true`

) if node`i`

represents a biconnected component. Otherwise, the value is equal to and logical`0`

(`false`

).`tree.Nodes.ComponentIndex(i)`

— Index indicating the component represented by node`i`

. The value is zero if node`i`

represents a cut vertex.`tree.Nodes.CutVertexIndex(i)`

— Index indicating the cut vertex represented by node`i`

. The value is zero if node`i`

represents a biconnected component.

`ind`

— Node indices

vector

Node indices, returned as a numeric vector. `ind(i)`

is
the node in the output graph `tree`

that represents node
`i`

in the input graph `G`

:

If node

`i`

is a cut vertex in graph`G`

, then`ind(i)`

is the associated node in`tree`

.If node

`i`

is not a cut vertex, but belongs to one of the biconnected components in graph`G`

, then`ind(i)`

is the node in`tree`

representing that biconnected component.If node

`i`

is an isolated node in graph`G`

, then`ind(i)`

is zero.

## More About

### Biconnected Components

A biconnected component of a graph is a maximally biconnected subgraph. A graph is biconnected if it does not contain any cut vertices.

Decomposing a graph into its biconnected components helps to measure how
well-connected the graph is. You can decompose any connected graph into a tree of
biconnected components, called the *block-cut tree*. The blocks
in the tree are attached at shared vertices, which are the cut vertices.

The illustration depicts:

(a) An undirected graph with 11 nodes.

(b) Five biconnected components of the graph, with the cut vertices of the original graph colored for each component to which they belong.

(c) Block-cut tree of the graph, which contains a node for each biconnected component (as large circles) and a node for each cut vertex (as smaller multicolored circles). In the block-cut tree, an edge connects each cut vertex to each component to which it belongs.

### Cut Vertices

Also known as *articulation points*, cut
vertices are graph nodes whose removal increases the number of connected components.
In the previous illustration, the cut vertices are those nodes with more than one
color: nodes 4, 6, and 7.

## Version History

**Introduced in R2016b**

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