The purpose is to light on LexBFS and Chordal Graphs Relation, Introductory

1. CORDAL GRAPH
Identification via Lex
BFS
Nazli Temur

2. Cordal Graphs [Triangulated Graphs]
• Definition. An undirected graph G is called Cordal-triangulated
if every cycle of length strictly greater than 3 possesses a
chord, that is, an edge joining two nonconsecutive vertices of
the cycle.
• For a graph G on n vertices, the following conditions
are equivalent:
• 1. G is chordal.
• 2.G has a perfect elimination ordering.
• 3. If H is any induced subgraph of G and S is a
vertex separator of H of minimal size, S’s vertices
induce a complete subgraph of G.

3. Perfect Elimination
Ordering
A graph G on n vertices is said to have
a perfect elimination ordering if and only
if there is an ordering {v1, . . . vn} of G’s
vertices, such that each vi is simplicial in
the subgraph induced by the vertices
{v1, . . . vi}.
As an example, the graph above has a perfect elimination ordering, witnessed by the
ordering (2, 1, 3, 4) of its vertices.

4. // Simplicial
• Definition. In a graph G, a vertex v is called simplicial if and only if the subgraph
(adjacency set) of G induced by the vertex set {v} ∪ N (v) is a complete graph. // a clique
(not necessarily maximal)]
• For example, in the graph below, vertex 3 is simplicial, while vertex 4 is not:
Vertex 3 Vertex4

5. Cordal Graphs [Triangulated Graphs]
• Definition. An undirected graph G is called Cordal-triangulated
if every cycle of length strictly greater than 3 possesses a
chord, that is, an edge joining two nonconsecutive vertices of
the cycle.
• For a graph G on n vertices, the following conditions
are equivalent:
• 1. G is chordal.
• 2.G has a perfect elimination ordering.
• 3. If H is any induced subgraph of G and S is a
vertex separator of H of minimal size, S’s vertices
induce a complete subgraph of G.

6. // Vertex Seperator
In graph theory, a vertex subset S subset V is a vertex separator for
nonadjacent vertices a and b if the removal of S from the graph
separates a and b into distinct connected components.

7. Note : A graph with no or one nodes
is connected.
Sconnected component 2
connected component 1
3rd Condition
If H is any induced subgraph of G and
S is a vertex separator of H of minimal
size, S’s vertices induce a complete
subgraph of G.
//induced subgraph is complete;

8. Cordal Graph Identification with LEX BFS
Rose, Lueker & Tarjan (1976) (see also Habib et al. 2000) show that
a perfect elimination ordering of a chordal graph may be found
efficiently using an algorithm known as lexicographic breadth-first
search.

9. LEX BFS on Cordal Graph
Sigma(a) a N’(a)-adj Partitions
L={2431}
1
2
3
4
5
• Vertices list is going be partitioned into two set P (Adj) and P’ (Non-Adj).
• The next vertex will be chosen such that it has lexicographically largest label.
• P’ is always located to the left most side of the Partitions List and will be processed
first.
• Any visited node will be removed from L=list.
• LexBFS terminates since all vertices have been visited.

10. Sigma(a) a N’(a)-adj Partitions
L={2431}
1 2 {1,4} {1,4} {3}
2
3
4
5
LEX BFS on Cordal Graph
2 is Removed
• Vertices list is going be partitioned into two set P (Adj) and P’ (Non-Adj).
• The next vertex will be chosen such that it has lexicographically largest label.
• P’ is always located to the left most side of the Partitions List and will be processed
first.
• Any visited node will be removed from L=list.
• LexBFS terminates since all vertices have been visited.

11. Sigma(a) a N’(a)-adj Partitions
L={2431}
1 2 {1,4} {1,4} {3}
2 1 {4,3} {4} {3}
3
4
5
1&2 is Removed
LEX BFS on Cordal Graph
• Vertices list is going be partitioned into two set P (Adj) and P’ (Non-Adj).
• The next vertex will be chosen such that it has lexicographically largest label.
• P’ is always located to the left most side of the Partitions List and will be processed
first.
• Any visited node will be removed from L=list.
• LexBFS terminates since all vertices have been visited.

12. Sigma(a) a N’(a)-adj Partitions
L={2431}
1 2 {1,4} {1,4} {3}
2 1 {4,3} {4} {3}
3 4 {3} {3}{}
4
5
LEX BFS on Cordal Graph
1&2&4 are Removed
• Vertices list is going be partitioned into two set P (Adj) and P’ (Non-Adj).
• The next vertex will be chosen such that it has lexicographically largest label.
• P’ is always located to the left most side of the Partitions List and will be processed
first.
• Any visited node will be removed from L=list.
• LexBFS terminates since all vertices have been visited.

13. Sigma(a) a N’(a)-adj Partitions
L={2431}
1 2 {1,4} {1,4} {3}
2 1 {4,3} {3}{4}
3 4 {3} {3}{}
4 3 {} {}
5 Terminate
ALL is Removed
LEX BFS on Cordal Graph
Sigma(a)=(2,1,4,3)
• Vertices list is going be partitioned into two set P (Adj) and P’ (Non-Adj).
• The next vertex will be chosen such that it has lexicographically largest label.
• P’ is always located to the left most side of the Partitions List and will be processed
first.
• Any visited node will be removed from L=list.
• LexBFS terminates since all vertices have been visited.

14. Conclusion
• Choral Graph and its induced subgraphs admits a perfect elimination ordering.
• Chordal graphes has vertex separators that separates the graph into distinct
connected components by removal their.
• LexBFS Algorithm is a polynomial algorithm that can give the answer of the
identity of a graph is chordal or not.
• There can be multiple perfect elimination ordering in a graph, Lex BFS can give
multiple ordering as a part of start not.
• However we need to know which percentage of perfect elimination orders can
be observed by Lex BFS.
• Still the labelling function of Lex BFS is not clear.

15. THANKS !