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On chordal and dually chordal graphs and their tree ...

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On chordal and dually chordal graphs and their tree representations Pablo De Caria CONICET/ Departamento de Matem´ atica, Universidad Nacional de La Plata Koper, Slovenia, October 2014 Pablo De Caria CONICET/ Departamento de Matem´ atica, Universidad Nacional de La Plata On chordal and dually chordal graphs and their tree representations
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Page 1: On chordal and dually chordal graphs and their tree ...

On chordal and dually chordal graphs andtheir tree representations

Pablo De Caria

CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

Koper, Slovenia, October 2014

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 2: On chordal and dually chordal graphs and their tree ...

Chordal graphsDefinition 1A chord is an edge connecting two nonconsecutive vertices of acycle.Definition 2A graph G is chordal if every cycle of length four or more in G hasa chord.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 3: On chordal and dually chordal graphs and their tree ...

Chordal graphsDefinition 1A chord is an edge connecting two nonconsecutive vertices of acycle.Definition 2A graph G is chordal if every cycle of length four or more in G hasa chord.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 4: On chordal and dually chordal graphs and their tree ...

Chordal graphsDefinition 1A chord is an edge connecting two nonconsecutive vertices of acycle.Definition 2A graph G is chordal if every cycle of length four or more in G hasa chord.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 5: On chordal and dually chordal graphs and their tree ...

Chordal graphsDefinition 1A chord is an edge connecting two nonconsecutive vertices of acycle.Definition 2A graph G is chordal if every cycle of length four or more in G hasa chord.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 6: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 7: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 8: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 9: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 10: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 11: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 12: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 13: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 14: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 15: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 16: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 17: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 18: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 19: On chordal and dually chordal graphs and their tree ...

Simplicial vertexIts neighborhood is a clique (maximal set of pairwise adjacentvertices).

Perfect elimination orderingAn ordering v1v2...vn of the vertices of the graph such that vi issimplicial in G [vi , ..., vn] for all i , 1 ≤ i ≤ n.

TheoremA graph G is chordal if and only if it has a perfect eliminationordering.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 20: On chordal and dually chordal graphs and their tree ...

Intersection graphThe intersection graph of the family F , or L(F), has F as vertexset and F1 and F2 are adjacent in L(F) if and only if F1 ∩ F2 6= ∅.

Another characterizationA graph is chordal if and only if it is the intersection graph of afamily of subtrees of a tree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 21: On chordal and dually chordal graphs and their tree ...

Intersection graphThe intersection graph of the family F , or L(F), has F as vertexset and F1 and F2 are adjacent in L(F) if and only if F1 ∩ F2 6= ∅.

Another characterizationA graph is chordal if and only if it is the intersection graph of afamily of subtrees of a tree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 22: On chordal and dually chordal graphs and their tree ...

Intersection graphThe intersection graph of the family F , or L(F), has F as vertexset and F1 and F2 are adjacent in L(F) if and only if F1 ∩ F2 6= ∅.

Another characterizationA graph is chordal if and only if it is the intersection graph of afamily of subtrees of a tree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 23: On chordal and dually chordal graphs and their tree ...

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 24: On chordal and dually chordal graphs and their tree ...

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 25: On chordal and dually chordal graphs and their tree ...

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 26: On chordal and dually chordal graphs and their tree ...

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 27: On chordal and dually chordal graphs and their tree ...

There are infinitely many representations for a chordal graph.

We can reduce the number of vertices in the representation bycontracting every edge uv of the tree such that every subtree containingu also contains v .

When no more contractions are possible, the family Fv , for v ∈ V (T ), ofsubtrees that contain v corresponds to pairwise adjacent vertices of thegraph and no Fv contains another.

ConclusionThere is a correspondence between the vertices of T and the cliques of G .

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 28: On chordal and dually chordal graphs and their tree ...

There are infinitely many representations for a chordal graph.

We can reduce the number of vertices in the representation bycontracting every edge uv of the tree such that every subtree containingu also contains v .

When no more contractions are possible, the family Fv , for v ∈ V (T ), ofsubtrees that contain v corresponds to pairwise adjacent vertices of thegraph and no Fv contains another.

ConclusionThere is a correspondence between the vertices of T and the cliques of G .

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 29: On chordal and dually chordal graphs and their tree ...

There are infinitely many representations for a chordal graph.

We can reduce the number of vertices in the representation bycontracting every edge uv of the tree such that every subtree containingu also contains v .

When no more contractions are possible, the family Fv , for v ∈ V (T ), ofsubtrees that contain v corresponds to pairwise adjacent vertices of thegraph and no Fv contains another.

ConclusionThere is a correspondence between the vertices of T and the cliques of G .

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 30: On chordal and dually chordal graphs and their tree ...

There are infinitely many representations for a chordal graph.

We can reduce the number of vertices in the representation bycontracting every edge uv of the tree such that every subtree containingu also contains v .

6

1

When no more contractions are possible, the family Fv , for v ∈ V (T ), ofsubtrees that contain v corresponds to pairwise adjacent vertices of thegraph and no Fv contains another.

ConclusionThere is a correspondence between the vertices of T and the cliques of G .

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 31: On chordal and dually chordal graphs and their tree ...

There are infinitely many representations for a chordal graph.

We can reduce the number of vertices in the representation bycontracting every edge uv of the tree such that every subtree containingu also contains v .

6

1

When no more contractions are possible, the family Fv , for v ∈ V (T ), ofsubtrees that contain v corresponds to pairwise adjacent vertices of thegraph and no Fv contains another.

ConclusionThere is a correspondence between the vertices of T and the cliques of G .

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 32: On chordal and dually chordal graphs and their tree ...

The clique treeA clique tree of G is a tree T whose vertices are the cliques of Gand such that, for every v ∈ V (G ), the set Cv of cliques containingv induces a subtree in T .

CharacterizationA graph is chordal if and only if it has a clique tree.

6

1

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 33: On chordal and dually chordal graphs and their tree ...

The clique treeA clique tree of G is a tree T whose vertices are the cliques of Gand such that, for every v ∈ V (G ), the set Cv of cliques containingv induces a subtree in T .

CharacterizationA graph is chordal if and only if it has a clique tree.

6

1

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 34: On chordal and dually chordal graphs and their tree ...

The clique treeA clique tree of G is a tree T whose vertices are the cliques of Gand such that, for every v ∈ V (G ), the set Cv of cliques containingv induces a subtree in T .

CharacterizationA graph is chordal if and only if it has a clique tree.

6

1

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 35: On chordal and dually chordal graphs and their tree ...

The clique treeA clique tree of G is a tree T whose vertices are the cliques of Gand such that, for every v ∈ V (G ), the set Cv of cliques containingv induces a subtree in T .

CharacterizationA graph is chordal if and only if it has a clique tree.

C2

C3

C4 C6

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 36: On chordal and dually chordal graphs and their tree ...

Clique graphsThe clique graph of a graph G , denoted by K (G ), is theintersection graph of the cliques of G .

PropertyFor G chordal and connected, any clique tree of G is a spanningtree of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 37: On chordal and dually chordal graphs and their tree ...

Clique graphsThe clique graph of a graph G , denoted by K (G ), is theintersection graph of the cliques of G .

PropertyFor G chordal and connected, any clique tree of G is a spanningtree of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 38: On chordal and dually chordal graphs and their tree ...

Clique graphsThe clique graph of a graph G , denoted by K (G ), is theintersection graph of the cliques of G .

PropertyFor G chordal and connected, any clique tree of G is a spanningtree of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 39: On chordal and dually chordal graphs and their tree ...

Clique graphsThe clique graph of a graph G , denoted by K (G ), is theintersection graph of the cliques of G .

PropertyFor G chordal and connected, any clique tree of G is a spanningtree of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 40: On chordal and dually chordal graphs and their tree ...

PropertyLet T be a clique tree and C be a clique of G . Then the cliques ofG that intersect C induce a subtree in T .

In other words, every clique tree of G verifies that every closedneighborhood of K (G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 41: On chordal and dually chordal graphs and their tree ...

PropertyLet T be a clique tree and C be a clique of G . Then the cliques ofG that intersect C induce a subtree in T .

In other words, every clique tree of G verifies that every closedneighborhood of K (G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 42: On chordal and dually chordal graphs and their tree ...

PropertyLet T be a clique tree and C be a clique of G . Then the cliques ofG that intersect C induce a subtree in T .

In other words, every clique tree of G verifies that every closedneighborhood of K (G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 43: On chordal and dually chordal graphs and their tree ...

PropertyLet T be a clique tree and C be a clique of G . Then the cliques ofG that intersect C induce a subtree in T .

In other words, every clique tree of G verifies that every closedneighborhood of K (G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 44: On chordal and dually chordal graphs and their tree ...

PropertyLet T be a clique tree and C be a clique of G . Then the cliques ofG that intersect C induce a subtree in T .

In other words, every clique tree of G verifies that every closedneighborhood of K (G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 45: On chordal and dually chordal graphs and their tree ...

PropertyLet T be a clique tree and C be a clique of G . Then the cliques ofG that intersect C induce a subtree in T .

In other words, every clique tree of G verifies that every closedneighborhood of K (G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 46: On chordal and dually chordal graphs and their tree ...

PropertyLet T be a clique tree and C be a clique of G . Then the cliques ofG that intersect C induce a subtree in T .

In other words, every clique tree of G verifies that every closedneighborhood of K (G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 47: On chordal and dually chordal graphs and their tree ...

Dually chordal graphs

A graph is dually chordal if it is the clique graph of a chordal graph.

Compatible tree

A compatible tree of a graph G is a tree T such that every closedneighborhood of G induces a subtree in T .

Theorem

Every dually chordal graph has a compatible tree.

What is more, a graph is dually chordal if and only if it has acompatible tree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 48: On chordal and dually chordal graphs and their tree ...

Dually chordal graphs

A graph is dually chordal if it is the clique graph of a chordal graph.

Compatible tree

A compatible tree of a graph G is a tree T such that every closedneighborhood of G induces a subtree in T .

Theorem

Every dually chordal graph has a compatible tree.

What is more, a graph is dually chordal if and only if it has acompatible tree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 49: On chordal and dually chordal graphs and their tree ...

Dually chordal graphs

A graph is dually chordal if it is the clique graph of a chordal graph.

Compatible tree

A compatible tree of a graph G is a tree T such that every closedneighborhood of G induces a subtree in T .

Theorem

Every dually chordal graph has a compatible tree.

What is more, a graph is dually chordal if and only if it has acompatible tree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 50: On chordal and dually chordal graphs and their tree ...

Why dually chordal graphs?Proposition

A tree T is compatible with G if and only if every clique induces asubtree in T .

Idea of proof: C =⋂

v∈C

N[v ] and N[v ] =⋃

v∈C

C

Dual familyThe dual of F is the family DF = {Dv}v∈

⋃F∈F

F , where

Dv = {F ∈ F : v ∈ F}.

ResultDefine C(G ) as the family of cliques of a graph G .

I G is dually chordal if and only if there exists a tree such that everymember of C(G ) induces a subtree.

I G is chordal if and only if there exists a tree such that every memberof DC(G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 51: On chordal and dually chordal graphs and their tree ...

Why dually chordal graphs?Proposition

A tree T is compatible with G if and only if every clique induces asubtree in T .

Idea of proof: C =⋂

v∈C

N[v ] and N[v ] =⋃

v∈C

C

Dual familyThe dual of F is the family DF = {Dv}v∈

⋃F∈F

F , where

Dv = {F ∈ F : v ∈ F}.

ResultDefine C(G ) as the family of cliques of a graph G .

I G is dually chordal if and only if there exists a tree such that everymember of C(G ) induces a subtree.

I G is chordal if and only if there exists a tree such that every memberof DC(G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 52: On chordal and dually chordal graphs and their tree ...

Why dually chordal graphs?Proposition

A tree T is compatible with G if and only if every clique induces asubtree in T .

Idea of proof: C =⋂

v∈C

N[v ] and N[v ] =⋃

v∈C

C

Dual familyThe dual of F is the family DF = {Dv}v∈

⋃F∈F

F , where

Dv = {F ∈ F : v ∈ F}.

ResultDefine C(G ) as the family of cliques of a graph G .

I G is dually chordal if and only if there exists a tree such that everymember of C(G ) induces a subtree.

I G is chordal if and only if there exists a tree such that every memberof DC(G ) induces a subtree.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 53: On chordal and dually chordal graphs and their tree ...

Property: Every clique tree of a graph G is a compatible tree ofK (G ).

Is the converse true?

A chordal graph G is said to be basic chordal if the clique trees ofG are exactly the compatible trees of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 54: On chordal and dually chordal graphs and their tree ...

Property: Every clique tree of a graph G is a compatible tree ofK (G ).

Is the converse true?

A chordal graph G is said to be basic chordal if the clique trees ofG are exactly the compatible trees of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 55: On chordal and dually chordal graphs and their tree ...

Property: Every clique tree of a graph G is a compatible tree ofK (G ).

Is the converse true?

A chordal graph G is said to be basic chordal if the clique trees ofG are exactly the compatible trees of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 56: On chordal and dually chordal graphs and their tree ...

Property: Every clique tree of a graph G is a compatible tree ofK (G ).

Is the converse true?

A chordal graph G is said to be basic chordal if the clique trees ofG are exactly the compatible trees of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 57: On chordal and dually chordal graphs and their tree ...

Property: Every clique tree of a graph G is a compatible tree ofK (G ).

Is the converse true?

A chordal graph G is said to be basic chordal if the clique trees ofG are exactly the compatible trees of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 58: On chordal and dually chordal graphs and their tree ...

Property: Every clique tree of a graph G is a compatible tree ofK (G ).

Is the converse true?

A chordal graph G is said to be basic chordal if the clique trees ofG are exactly the compatible trees of K (G ).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 59: On chordal and dually chordal graphs and their tree ...

G is basic chordal if the compatible trees of K (G ) are exactly theclique trees of G .

Let S(G ) be the family of minimal vertex separators of G and letS ∈ S(G ).

CS : Cliques that contain S

BS : Consists of every C such that C ∩ D 6= ∅ for D ∈ C(G ) suchthat D ∩ S 6= ∅.

S = {2, 5}

CS = {C2,C3}

BS = {C2,C3,C4}

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 60: On chordal and dually chordal graphs and their tree ...

G is basic chordal if the compatible trees of K (G ) are exactly theclique trees of G .

Let S(G ) be the family of minimal vertex separators of G and letS ∈ S(G ).

CS : Cliques that contain S

BS : Consists of every C such that C ∩ D 6= ∅ for D ∈ C(G ) suchthat D ∩ S 6= ∅.

S = {2, 5}

CS = {C2,C3}

BS = {C2,C3,C4}

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 61: On chordal and dually chordal graphs and their tree ...

G is basic chordal if the compatible trees of K (G ) are exactly theclique trees of G .

Let S(G ) be the family of minimal vertex separators of G and letS ∈ S(G ).

CS : Cliques that contain S

BS : Consists of every C such that C ∩ D 6= ∅ for D ∈ C(G ) suchthat D ∩ S 6= ∅.

S = {2, 5}

CS = {C2,C3}

BS = {C2,C3,C4}

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 62: On chordal and dually chordal graphs and their tree ...

Theorem

A graph G is basic chordal iff

BS = CS for all S ∈ S(G ).

S BS CS BS = CS

{2, 3} {C1,C3} {C1,C3} X{2, 5} {C2,C3} {C2,C3} X{3, 5} {C3,C4} {C3,C4} X{2} {C1,C2,C3,C5} {C1,C2,C3,C5} X{3} {C1,C3,C4,C6} {C1,C3,C4,C6} X{5} {C2,C3,C4,C7} {C2,C3,C4,C7} X

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 63: On chordal and dually chordal graphs and their tree ...

Theorem

A graph G is basic chordal iff

BS = CS for all S ∈ S(G ).

S BS CS BS = CS

{2, 3} {C1,C3} {C1,C3} X{2, 5} {C2,C3} {C2,C3} X{3, 5} {C3,C4} {C3,C4} X{2} {C1,C2,C3,C5} {C1,C2,C3,C5} X{3} {C1,C3,C4,C6} {C1,C3,C4,C6} X{5} {C2,C3,C4,C7} {C2,C3,C4,C7} X

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 64: On chordal and dually chordal graphs and their tree ...

Theorem

A graph G is basic chordal iff

BS = CS for all S ∈ S(G ).

S BS CS BS = CS

{2, 3} {C1,C3} {C1,C3} X{2, 5} {C2,C3} {C2,C3} X{3, 5} {C3,C4} {C3,C4} X{2} {C1,C2,C3,C5} {C1,C2,C3,C5} X{3} {C1,C3,C4,C6} {C1,C3,C4,C6} X{5} {C2,C3,C4,C7} {C2,C3,C4,C7} X

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 65: On chordal and dually chordal graphs and their tree ...

If G is chordal,

SC(G ): Sets that induce a subtree of every clique tree of G .

Example: The members of DC(G ).

If G is dually chordal,

SDC(G ): Sets that induce a subtree of every compatible tree of G .

Example: Cliques, closed neighborhoods and minimal vertexseparators.

Theorem

A chordal graph G is basic chordal if and only ifSC(G ) = SDC(K (G )).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 66: On chordal and dually chordal graphs and their tree ...

If G is chordal,

SC(G ): Sets that induce a subtree of every clique tree of G .

Example: The members of DC(G ).

If G is dually chordal,

SDC(G ): Sets that induce a subtree of every compatible tree of G .

Example: Cliques, closed neighborhoods and minimal vertexseparators.

Theorem

A chordal graph G is basic chordal if and only ifSC(G ) = SDC(K (G )).

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 67: On chordal and dually chordal graphs and their tree ...

Connected unions⋃F∈F

F is connected if L(F) is connected.

Connected unions of members of SC(G )/SDC(G ) are inSC(G )/SDC(G ).

Families closed under connected unions are characterized by theexistence of a basis.

A basis of F is a minimal subfamily B such that every F ∈ F with|F | ≥ 2 can be expressed as the connected union of members of B.

Example: A = {1, 2, 3, 4}{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} form the basis of the powerset of A.

{1, 2, 3, 4} = {1, 2} ∪ {2, 3} ∪ {3, 4}

A basis is always unique and consists of the sets that cannot beexpressed as connected unions of others.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 68: On chordal and dually chordal graphs and their tree ...

Connected unions⋃F∈F

F is connected if L(F) is connected.

Connected unions of members of SC(G )/SDC(G ) are inSC(G )/SDC(G ).

Families closed under connected unions are characterized by theexistence of a basis.

A basis of F is a minimal subfamily B such that every F ∈ F with|F | ≥ 2 can be expressed as the connected union of members of B.

Example: A = {1, 2, 3, 4}{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} form the basis of the powerset of A.

{1, 2, 3, 4} = {1, 2} ∪ {2, 3} ∪ {3, 4}

A basis is always unique and consists of the sets that cannot beexpressed as connected unions of others.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 69: On chordal and dually chordal graphs and their tree ...

Connected unions⋃F∈F

F is connected if L(F) is connected.

Connected unions of members of SC(G )/SDC(G ) are inSC(G )/SDC(G ).

Families closed under connected unions are characterized by theexistence of a basis.

A basis of F is a minimal subfamily B such that every F ∈ F with|F | ≥ 2 can be expressed as the connected union of members of B.

Example: A = {1, 2, 3, 4}{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} form the basis of the powerset of A.

{1, 2, 3, 4} = {1, 2} ∪ {2, 3} ∪ {3, 4}

A basis is always unique and consists of the sets that cannot beexpressed as connected unions of others.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 70: On chordal and dually chordal graphs and their tree ...

Connected unions⋃F∈F

F is connected if L(F) is connected.

Connected unions of members of SC(G )/SDC(G ) are inSC(G )/SDC(G ).

Families closed under connected unions are characterized by theexistence of a basis.

A basis of F is a minimal subfamily B such that every F ∈ F with|F | ≥ 2 can be expressed as the connected union of members of B.

Example: A = {1, 2, 3, 4}{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} form the basis of the powerset of A.

{1, 2, 3, 4} = {1, 2} ∪ {2, 3} ∪ {3, 4}

A basis is always unique and consists of the sets that cannot beexpressed as connected unions of others.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 71: On chordal and dually chordal graphs and their tree ...

Why basic chordal graphs?G is basic chordal if and only if SC(G ) and SDC(K (G )) have thesame basis.

What is the basis of SC(G ) for G chordal?

The basis consists of the sets CS , S ∈ S(G ).

What is the basis of SDC(K (G ))?

It consists of the sets BS , S ∈ S(G ).

How can the basis for a dually chordal graph be obtained withoutusing chordal graphs?

Take T compatible and, for every uv ∈ E (T ), Consider⋂{u,v}⊆C

C =⋂

{u,v}⊆N[w ]

N[w ].

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 72: On chordal and dually chordal graphs and their tree ...

Why basic chordal graphs?G is basic chordal if and only if SC(G ) and SDC(K (G )) have thesame basis.

What is the basis of SC(G ) for G chordal?

The basis consists of the sets CS , S ∈ S(G ).

What is the basis of SDC(K (G ))?

It consists of the sets BS , S ∈ S(G ).

How can the basis for a dually chordal graph be obtained withoutusing chordal graphs?

Take T compatible and, for every uv ∈ E (T ), Consider⋂{u,v}⊆C

C =⋂

{u,v}⊆N[w ]

N[w ].

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 73: On chordal and dually chordal graphs and their tree ...

Why basic chordal graphs?G is basic chordal if and only if SC(G ) and SDC(K (G )) have thesame basis.

What is the basis of SC(G ) for G chordal?

The basis consists of the sets CS , S ∈ S(G ).

What is the basis of SDC(K (G ))?

It consists of the sets BS , S ∈ S(G ).

How can the basis for a dually chordal graph be obtained withoutusing chordal graphs?

Take T compatible and, for every uv ∈ E (T ), Consider⋂{u,v}⊆C

C =⋂

{u,v}⊆N[w ]

N[w ].

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 74: On chordal and dually chordal graphs and their tree ...

The only clique containing C1C2, is {C1,C2,C3,C4}. But it is nota basic set for the graph at left.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 75: On chordal and dually chordal graphs and their tree ...

Clique graphs of basic chordal graphsSeparating familyF is separating if for every vertex v ,

⋂v∈F

F = {v}.

Theorem

I K(BASIC CHORDAL) = DUALLY CHORDALI For G dually chordal, the basic chordal graphs with G as a clique graph

are of the form L(F), where F is separating, F ⊆ SDC(G) and F coversall the edges of G .

A = {C1, C2, C3, C4}, B = {C2, C3, C4, C5}, C = {C1}, D = {C2}, E = {C3},F = {C4}, G = {C5}

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 76: On chordal and dually chordal graphs and their tree ...

Clique graphs of basic chordal graphsSeparating familyF is separating if for every vertex v ,

⋂v∈F

F = {v}.

Theorem

I K(BASIC CHORDAL) = DUALLY CHORDALI For G dually chordal, the basic chordal graphs with G as a clique graph

are of the form L(F), where F is separating, F ⊆ SDC(G) and F coversall the edges of G .

A = {C1, C2, C3, C4}, B = {C2, C3, C4, C5}, C = {C1}, D = {C2}, E = {C3},F = {C4}, G = {C5}

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 77: On chordal and dually chordal graphs and their tree ...

Clique graphs of basic chordal graphsSeparating familyF is separating if for every vertex v ,

⋂v∈F

F = {v}.

Theorem

I K(BASIC CHORDAL) = DUALLY CHORDALI For G dually chordal, the basic chordal graphs with G as a clique graph

are of the form L(F), where F is separating, F ⊆ SDC(G) and F coversall the edges of G .

A = {C1, C2, C3, C4}, B = {C2, C3, C4, C5}, C = {C1}, D = {C2}, E = {C3},F = {C4}, G = {C5}

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 78: On chordal and dually chordal graphs and their tree ...

Clique graphs of basic chordal graphsSeparating familyF is separating if for every vertex v ,

⋂v∈F

F = {v}.

Theorem

I K(BASIC CHORDAL) = DUALLY CHORDALI For G dually chordal, the basic chordal graphs with G as a clique graph

are of the form L(F), where F is separating, F ⊆ SDC(G) and F coversall the edges of G .

A = {C1, C2, C3, C4}, B = {C2, C3, C4, C5}, C = {C1}, D = {C2}, E = {C3},F = {C4}, G = {C5}

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 79: On chordal and dually chordal graphs and their tree ...

Clique graphs of basic chordal graphsSeparating familyF is separating if for every vertex v ,

⋂v∈F

F = {v}.

Theorem

I K(BASIC CHORDAL) = DUALLY CHORDALI For G dually chordal, the basic chordal graphs with G as a clique graph

are of the form L(F), where F is separating, F ⊆ SDC(G) and F coversall the edges of G .

A = {C1, C2, C3, C4}, B = {C2, C3, C4, C5}, C = {C1}, D = {C2}, E = {C3},F = {C4}, G = {C5}

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 80: On chordal and dually chordal graphs and their tree ...

Applications

Leafage of a chordal graph G : It is the minimum number ofleaves of a clique tree of G .

Can be found polinomially with an algorithm developed by M.Habib and J. Stacho.

Dual leafage of a graph G : Minimum number of leaves of acompatible tree of G .

The dual leafage of G is equal to the leafage of any basic chordalgraph H with K (H) = G .

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 81: On chordal and dually chordal graphs and their tree ...

Applications

Leafage of a chordal graph G : It is the minimum number ofleaves of a clique tree of G .

Can be found polinomially with an algorithm developed by M.Habib and J. Stacho.

Dual leafage of a graph G : Minimum number of leaves of acompatible tree of G .

The dual leafage of G is equal to the leafage of any basic chordalgraph H with K (H) = G .

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 82: On chordal and dually chordal graphs and their tree ...

ProblemLet G be a dually chordal graph, |V (G )| ≥ 3 and A ⊆ V (G ). Isthere a compatible tree of G that has A as its set of leaves?

Necessary conditionEvery vertex of A is dominated by another in V (G ) \ A.

Let G ∗ be obtained from G by adding, for each v ∈ A, the vertexv∗ and the edge vv∗.

Theorem

If every vertex of A is dominated by another in V (G ) \ A, then Ghas a compatible tree with set of leaves equal to A if and only ifthe dual leafage of G ∗ equals |A|.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 83: On chordal and dually chordal graphs and their tree ...

ProblemLet G be a dually chordal graph, |V (G )| ≥ 3 and A ⊆ V (G ). Isthere a compatible tree of G that has A as its set of leaves?

Necessary conditionEvery vertex of A is dominated by another in V (G ) \ A.

Let G ∗ be obtained from G by adding, for each v ∈ A, the vertexv∗ and the edge vv∗.

Theorem

If every vertex of A is dominated by another in V (G ) \ A, then Ghas a compatible tree with set of leaves equal to A if and only ifthe dual leafage of G ∗ equals |A|.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 84: On chordal and dually chordal graphs and their tree ...

ProblemLet G be a dually chordal graph, |V (G )| ≥ 3 and A ⊆ V (G ). Isthere a compatible tree of G that has A as its set of leaves?

Necessary conditionEvery vertex of A is dominated by another in V (G ) \ A.

Let G ∗ be obtained from G by adding, for each v ∈ A, the vertexv∗ and the edge vv∗.

Theorem

If every vertex of A is dominated by another in V (G ) \ A, then Ghas a compatible tree with set of leaves equal to A if and only ifthe dual leafage of G ∗ equals |A|.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 85: On chordal and dually chordal graphs and their tree ...

Problem

Given a family T on a set V of vertices. Is there a chordal graphwhose clique trees are exactly those of T ?

Theorem

I T is a clique tree of G if and only if CS induces a subtree of Tfor every S ∈ S(G ).

I The intersection graph of {CS}S∈S(G) ∪ {{C}}C∈C(G) has thesame clique trees as G .

Question

How to identify the sets CS when the graph is unknown and allwhat we know about it are its clique trees?

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 86: On chordal and dually chordal graphs and their tree ...

Problem

Given a family T on a set V of vertices. Is there a chordal graphwhose clique trees are exactly those of T ?

Theorem

I T is a clique tree of G if and only if CS induces a subtree of Tfor every S ∈ S(G ).

I The intersection graph of {CS}S∈S(G) ∪ {{C}}C∈C(G) has thesame clique trees as G .

Question

How to identify the sets CS when the graph is unknown and allwhat we know about it are its clique trees?

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 87: On chordal and dually chordal graphs and their tree ...

Problem

Given a family T on a set V of vertices. Is there a chordal graphwhose clique trees are exactly those of T ?

Theorem

I T is a clique tree of G if and only if CS induces a subtree of Tfor every S ∈ S(G ).

I The intersection graph of {CS}S∈S(G) ∪ {{C}}C∈C(G) has thesame clique trees as G .

Question

How to identify the sets CS when the graph is unknown and allwhat we know about it are its clique trees?

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 88: On chordal and dually chordal graphs and their tree ...

Definitions

T [a, b] : Vertices in the path of T from a to b.

T [a, b] =⋃

T∈TT [a, b].

TheoremLet G be chordal, T be its family of clique trees and T ∈ T . Then{CS}S∈S(G) = {T [C ,C ′] : CC ′ ∈ E (T )}.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 89: On chordal and dually chordal graphs and their tree ...

Definitions

T [a, b] : Vertices in the path of T from a to b.

T [a, b] =⋃

T∈TT [a, b].

TheoremLet G be chordal, T be its family of clique trees and T ∈ T . Then{CS}S∈S(G) = {T [C ,C ′] : CC ′ ∈ E (T )}.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 90: On chordal and dually chordal graphs and their tree ...

TheoremLet T be a family of trees on a set of vertices V , T ∈ T andF = {T [u, v ]}uv∈E(T ) ∪ {{v}}v∈V (T ). Then T is the family ofclique trees of a chordal graph if and only if L(F) is chordal andand has |T | clique trees.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 91: On chordal and dually chordal graphs and their tree ...

TheoremLet T be a family of trees on a set of vertices V , T ∈ T andF = {T [u, v ]}uv∈E(T ) ∪ {{v}}v∈V (T ). Then T is the family ofclique trees of a chordal graph if and only if L(F) is chordal andand has |T | clique trees.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations

Page 92: On chordal and dually chordal graphs and their tree ...

I Every set of compatible trees of some dually chordal graph isthe set of clique trees of some chordal graph.

I However, the converse is not true.

I Determining the complexity of recognizing families ofcompatible trees is an open problem.

Pablo De Caria CONICET/ Departamento de Matematica, Universidad Nacional de La Plata

On chordal and dually chordal graphs and their tree representations


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