1 | initial version |

Recall that a block graph is a graph in which every biconnected component is a clique.

A simple method to generate a connected block graph with m blocks of size k is to start from a random tree with m edges and then replace each edge by a clique of order k:

```
def RandomBlockGraph(m, k):
G = graphs.RandomTree(m+1)
E = G.edges(labels=0)
for u,v in E:
G.add_clique([u,v] + [G.add_vertex() for i in range(k-2)])
return G
```

2 | No.2 Revision |

Recall that a block graph is a graph in which every biconnected component is a clique.

A simple method to generate a connected block graph with m blocks of size k is to start from a random tree with m edges and then replace each edge by a clique of order k:

```
def RandomBlockGraph(m, k):
G = graphs.RandomTree(m+1)
E = G.edges(labels=0)
for u,v in E:
G.add_clique([u,v] + [G.add_vertex() for i in range(k-2)])
return G
```

The `add_clique`

method is a recent addition to sagemath. If you don't have it, you can do

```
def RandomBlockGraph(m, k):
import itertools
G = graphs.RandomTree(m+1)
E = G.edges(labels=0)
for u,v in E:
V = [u,v] + [G.add_vertex() for i in range(k-2)]
for x,y in itertools.combinations(V, 2):
G.add_edge(x, y)
return G
```

A call to `G.add_vertex()`

adds an isolated vertex to the graph. You then add the edges of the clique.

3 | No.3 Revision |

Recall that a block graph is a graph in which every biconnected component is a clique.

A simple method to generate a connected block graph with m blocks of size k is to start from a random tree with m edges and then replace each edge by a clique of order k:

```
def RandomBlockGraph(m, k):
G = graphs.RandomTree(m+1)
E = G.edges(labels=0)
for u,v in E:
G.add_clique([u,v] + [G.add_vertex() for i in range(k-2)])
return G
```

The `add_clique`

method is a recent addition to sagemath. If you don't have it, you can do

```
def RandomBlockGraph(m, k):
import itertools
G = graphs.RandomTree(m+1)
E = G.edges(labels=0)
for u,v in E:
V = [u,v] + [G.add_vertex() for i in range(k-2)]
for x,y in itertools.combinations(V, 2):
G.add_edge(x, y)
return G
```

A call to `G.add_vertex()`

adds an isolated vertex to the graph. You then add the edges of the clique.

*EDIT*: The method below starts with a random tree of order m. Then it creates a graph with one k-clique (block) per vertex of the tree. We connect blocks corresponding to end vertices of tree edges. To do so, we select one vertex per block and merge the two selected vertices. We use a disjoint set data structure to keep track of the merge operations and so get a unique identifier per set of merged vertices.

```
def RandomBlockGraph(m, k):
import itertools
import random
# We start with a random tree of order m
T = graphs.RandomTree(m)
# We create a block of size k per vertex of the tree
B = {u:[(u,i) for i in range(k)] for u in T}
# For each edge of the tree, we choose 1 vertex in each of the corresponding
# blocks and we merge them. We use a disjoint set data structure to keep a
# unique identifier per merged vertices
DS = DisjointSet([i for u in B for i in B[u]])
for u,v in T.edges(labels=0):
DS.union(random.choice(B[u]), random.choice(B[v]))
# We finally build the block graph
BG = Graph()
for u in B:
for x,y in itertools.combinations(B[u], 2):
BG.add_edge(DS.find(x), DS.find(y))
return BG
```

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