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Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Hyperbolicity and Other Tree-likenessParameters
Yaokun Wu (�n)- ykwu@sjtu.edu.cn
Department of Mathematics, Shanghai Jiao Tong University
USTC Workshop on Graph Theory and CombinatoricsMay 30, 2010
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Figure: Somewhere in FeiDong with those beautiful trees in ouryounger days...
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Outline
1 HyperbolicityBasic DefinitionRooted versus Unrooted0-Hyperbolic Graphs and Beyond
2 Easily Related Metric Graph Theory ConceptsEccentricityCartesian ProductBreadth Property
3 Chordality
4 Tree-length
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Graph Metric Space
We only consider simple, connected, unweighted, but notnecessarily finite graphs.
Any graph G together with the usual shortest-path metric on it,dG : V (G)× V (G) 7→ {0, 1, 2, . . .}, gives rise to a metric space.
We often use the shorthand xy for dG(x , y). Note that a pair ofvertices x and y form an edge if and only if xy = 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
A Function on Quartets
For any vertices x , y , u, v of a graph G, put δG(x , y , u, v) to bethe difference between the largest and the second largest ofthe following three quantities
xu + yv2
,xv + yu
2, and
xy + uv2
.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
δ(x , x , u, v) = 0
If x = y , then we have
xu + yv= xv + yu= xu + xv ≥ uv =xy + uv .
By symmetry, this means that δ(x , y , u, v) = 0 providedx , y , u, v are not 4 different vertices.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
δ(x , x , u, v) = 0
If x = y , then we have
xu + yv= xv + yu= xu + xv ≥ uv =xy + uv .
By symmetry, this means that δ(x , y , u, v) = 0 providedx , y , u, v are not 4 different vertices.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Tree and the Four-Point Condition (4PC)
ry
rur
v
rx ra rb
xu + yv = xv + yu = xy + uv + 2ab ⇒ δ(x , y , u, v) = 0
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Hyperbolicity
A graph G, viewed as a metric space, is δ-hyperbolicprovided for any vertices x , y , u, v in G it holdsδ(x , y , u, v) ≤ δ
and the (Gromov) hyperbolicity of G,denoted δ∗(G), is the minimum half integer δ such that G isδ-hyperbolic.
Note that it may happen δ∗(G) = ∞. But for a finite graphG, δ∗(G) is clearly finite and polynomial time computable.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Hyperbolicity
A graph G, viewed as a metric space, is δ-hyperbolicprovided for any vertices x , y , u, v in G it holdsδ(x , y , u, v) ≤ δ
and the (Gromov) hyperbolicity of G,denoted δ∗(G), is the minimum half integer δ such that G isδ-hyperbolic.
Note that it may happen δ∗(G) = ∞. But for a finite graphG, δ∗(G) is clearly finite and polynomial time computable.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Hyperbolicity
A graph G, viewed as a metric space, is δ-hyperbolicprovided for any vertices x , y , u, v in G it holdsδ(x , y , u, v) ≤ δ and the (Gromov) hyperbolicity of G,denoted δ∗(G), is the minimum half integer δ such that G isδ-hyperbolic.
Note that it may happen δ∗(G) = ∞. But for a finite graphG, δ∗(G) is clearly finite and polynomial time computable.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Hyperbolicity
A graph G, viewed as a metric space, is δ-hyperbolicprovided for any vertices x , y , u, v in G it holdsδ(x , y , u, v) ≤ δ and the (Gromov) hyperbolicity of G,denoted δ∗(G), is the minimum half integer δ such that G isδ-hyperbolic.Note that it may happen δ∗(G) = ∞. But for a finite graphG, δ∗(G) is clearly finite and polynomial time computable.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Tree Metric
Every tree has hyperbolicity zero.
A metric space is a real tree (Tits 1977) if and only if it ispath-connected and 0-hyperbolic.
The hyperbolicity of a graph is a way to measure the additivedistortion with which every four-points sub-metric of the givengraph metric embeds into a tree metric, namely it is atree-likeness parameter.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Tree Metric
Every tree has hyperbolicity zero.
A metric space is a real tree (Tits 1977) if and only if it ispath-connected and 0-hyperbolic.
The hyperbolicity of a graph is a way to measure the additivedistortion with which every four-points sub-metric of the givengraph metric embeds into a tree metric, namely it is atree-likeness parameter.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Tree Metric
Every tree has hyperbolicity zero.
A metric space is a real tree (Tits 1977) if and only if it ispath-connected and 0-hyperbolic.
The hyperbolicity of a graph is a way to measure the additivedistortion with which every four-points sub-metric of the givengraph metric embeds into a tree metric, namely it is atree-likeness parameter.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Weakly Geodesic Graphs
A weakly geodesic graph is a graph in which any two vertices atdistance two have exactly one common neighbor.
If G is not weakly geodesic, we can find x , y , u, v such thatxy = 2 and u, v are two different vertices lying in the geodesicsbetween x and y .This gives
xy + uv ≥ 3 > 2 = xu + yv = xv + yu
and hence δ∗(G) ≥ δ(x , y , u, v) ≥ 12 .
We now see that every 0-hyperbolic graph is weakly geodesic.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Weakly Geodesic Graphs
A weakly geodesic graph is a graph in which any two vertices atdistance two have exactly one common neighbor.
If G is not weakly geodesic, we can find x , y , u, v such thatxy = 2 and u, v are two different vertices lying in the geodesicsbetween x and y .
This gives
xy + uv ≥ 3 > 2 = xu + yv = xv + yu
and hence δ∗(G) ≥ δ(x , y , u, v) ≥ 12 .
We now see that every 0-hyperbolic graph is weakly geodesic.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Weakly Geodesic Graphs
A weakly geodesic graph is a graph in which any two vertices atdistance two have exactly one common neighbor.
If G is not weakly geodesic, we can find x , y , u, v such thatxy = 2 and u, v are two different vertices lying in the geodesicsbetween x and y .This gives
xy + uv ≥ 3 > 2 = xu + yv = xv + yu
and hence δ∗(G) ≥ δ(x , y , u, v) ≥ 12 .
We now see that every 0-hyperbolic graph is weakly geodesic.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Weakly Geodesic Graphs
A weakly geodesic graph is a graph in which any two vertices atdistance two have exactly one common neighbor.
If G is not weakly geodesic, we can find x , y , u, v such thatxy = 2 and u, v are two different vertices lying in the geodesicsbetween x and y .This gives
xy + uv ≥ 3 > 2 = xu + yv = xv + yu
and hence δ∗(G) ≥ δ(x , y , u, v) ≥ 12 .
We now see that every 0-hyperbolic graph is weakly geodesic.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Weakly Geodesic Graphs
A weakly geodesic graph is a graph in which any two vertices atdistance two have exactly one common neighbor.
If G is not weakly geodesic, we can find x , y , u, v such thatxy = 2 and u, v are two different vertices lying in the geodesicsbetween x and y .This gives
xy + uv ≥ 3 > 2 = xu + yv = xv + yu
and hence δ∗(G) ≥ δ(x , y , u, v) ≥ 12 .
We now see that every 0-hyperbolic graph is weakly geodesic.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Cycle
The hyperbolicity of the n-cycle Cn is bn4c −
12 if n is congruent
to 1 modulo 4 and is bn4c else.
In particular, Cn has nonzero hyperbolicity for n > 3. Note thatCn is weakly geodesic when n > 4.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Cycle
The hyperbolicity of the n-cycle Cn is bn4c −
12 if n is congruent
to 1 modulo 4 and is bn4c else.
In particular, Cn has nonzero hyperbolicity for n > 3. Note thatCn is weakly geodesic when n > 4.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Mikhail Gromov
The concept of hyperbolicity comes from the work of Gromov ingeometric group theory which encapsulates many of the globalfeatures of the geometry of complete, simply connectedmanifolds of negative curvature. – M. Bridson, A. Haefliger,Metric Spaces of Non-Positive Curvature, Springer, 1999.
It is incredible what Mikhail Gromov can do, just with thetriangle inequality. – Dennis Sullivan
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Basic Definition
Mikhail Gromov
The concept of hyperbolicity comes from the work of Gromov ingeometric group theory which encapsulates many of the globalfeatures of the geometry of complete, simply connectedmanifolds of negative curvature. – M. Bridson, A. Haefliger,Metric Spaces of Non-Positive Curvature, Springer, 1999.
It is incredible what Mikhail Gromov can do, just with thetriangle inequality. – Dennis Sullivan
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Rooted versus Unrooted
Gromov Product
For any vertex a ∈ V (G), the Gromov product, also known asthe overlap function, of any two vertices x and y of G withrespect to a is equal to 1
2(xa + ya− xy) and is denoted by(x · y)a.
We say that G is δ-hyperbolic with respect to a ∈ V (G) if for anyx , y , v ∈ G it holds
(x · y)a ≥ min((x · v)a, (y · v)a)− δ.
It is easy to check that G is δ-hyperbolic if and only if G isδ-hyperbolic with respect to every vertex of G.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Rooted versus Unrooted
Gromov Product
For any vertex a ∈ V (G), the Gromov product, also known asthe overlap function, of any two vertices x and y of G withrespect to a is equal to 1
2(xa + ya− xy) and is denoted by(x · y)a.
We say that G is δ-hyperbolic with respect to a ∈ V (G) if for anyx , y , v ∈ G it holds
(x · y)a ≥ min((x · v)a, (y · v)a)− δ.
It is easy to check that G is δ-hyperbolic if and only if G isδ-hyperbolic with respect to every vertex of G.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Rooted versus Unrooted
Gromov Product
For any vertex a ∈ V (G), the Gromov product, also known asthe overlap function, of any two vertices x and y of G withrespect to a is equal to 1
2(xa + ya− xy) and is denoted by(x · y)a.
We say that G is δ-hyperbolic with respect to a ∈ V (G) if for anyx , y , v ∈ G it holds
(x · y)a ≥ min((x · v)a, (y · v)a)− δ.
It is easy to check that G is δ-hyperbolic if and only if G isδ-hyperbolic with respect to every vertex of G.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Rooted versus Unrooted
Farris Transform in Mathematics and Phylogenetics
Think of a as the ancestor of all species, take a very largenumber ρ, then ρ− (x · y)a is an estimate for the number ofyears that have passed since the last common ancestor of xand y existed.
4 A. Dress, K.T. Huber, and V. Moulton
a
xy
lca(x,y)
Figure 2: The dotted lines depict D(a,ρ)(x,y) which can be thought of as an estimate forthe number of years that have passed since lca(x,y) existed.
in view of
ρ+12
(DGC(x, y)−DGC(x, a)−DGC(y, a)) = ρ−DGC (a, lca(x, y)) =
ρ−DGC(a, v)−DGC(lca(x, y), v) = ρ0−D∗GC(lca(x, y), v) =
D∗GC(z, v)−D∗
GC(lca(x, y), v) = D∗GC (z, lca(x, y))
for both, z := x and z := y.Considerations like this may have led Farris and his collaborators in 1970 to de-
fine, for any dissimilarity D : X ×X → R : (x, y) 7→ xy := D(x, y), any element a ∈ Xrepresenting a putative outgroup, and any large positive real number ρ, the map
D(a,ρ) : X ×X → R,
later to be dubbed the Farris transform of D relative to a and ρ, by
D(a,ρ)(x, x) := 0,
for all x ∈ X , and by
D(a,ρ)(x, y) := ρ+12(xy− xa− ya),
for all x, y ∈ X with x 6= y, and to suggest these numbers D(a,ρ)(x, y) as likely estimates,up to a fixed proportionality factor, for the number of years that passed since lca(x, y)existed (cf. Figure 2).
This transformation, originally suggested by Farris et al. in [34, p. 181] (see also[32, p. 83, 5ff] and [33, p. 491]) was dubbed the Farris transform in [4] (see also [6]),and it was rediscovered in the context of phylogenetics in both, [11] and [41]. Intrigu-ingly, the same transformation has also come up in investigations of apparently ratherunrelated mathematical topics: For example, it was implicitly used in the study of hy-perbolic groups by Gromov [36, p. 158] (see also [10, p. 134]) and later also dubbed,in [17], the Gromov product. In disguise, it appears also in Distance Geometry where itis known as the covariance mapping (see [23, p. 56] for more details).
The Farris transform based at a sends the given (graph) metricdG to the map
Dρ,a(dG) : V (G)× V (G) → R : (x , y) 7→ ρ− (x · y)a.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Tree
A maximal connected subgraph without a cut vertex is called ablock.
The hyperbolicity of a graph is the maximum hyperbolicityof its blocks.
rg
re r f
r d
r c
ra rb-
rg
re rfrdrdrcrc
ra rb
B3
B2
B1
-
rB3
rdrB2
rcrB1
The block tree of G has the cut vertices and blocks of G as itsvertices and cB is an edge if c is a cut vertex inside the blockB.The block tree is always a tree.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Tree
A maximal connected subgraph without a cut vertex is called ablock.The hyperbolicity of a graph is the maximum hyperbolicityof its blocks.
rg
re r f
r d
r c
ra rb-
rg
re rfrdrdrcrc
ra rb
B3
B2
B1
-
rB3
rdrB2
rcrB1
The block tree of G has the cut vertices and blocks of G as itsvertices and cB is an edge if c is a cut vertex inside the blockB.The block tree is always a tree.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Tree
A maximal connected subgraph without a cut vertex is called ablock.The hyperbolicity of a graph is the maximum hyperbolicityof its blocks.
rg
re r f
r d
r c
ra rb-
rg
re rfrdrdrcrc
ra rb
B3
B2
B1
-
rB3
rdrB2
rcrB1
The block tree of G has the cut vertices and blocks of G as itsvertices and cB is an edge if c is a cut vertex inside the blockB.
The block tree is always a tree.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Tree
A maximal connected subgraph without a cut vertex is called ablock.The hyperbolicity of a graph is the maximum hyperbolicityof its blocks.
rg
re r f
r d
r c
ra rb-
rg
re rfrdrdrcrc
ra rb
B3
B2
B1
-
rB3
rdrB2
rcrB1
The block tree of G has the cut vertices and blocks of G as itsvertices and cB is an edge if c is a cut vertex inside the blockB.The block tree is always a tree.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Graphs = 0-Hyperbolic Graphs
Howorka (1979), Bandelt, Mulder (1986), Dress, Moulton, Steel(1997)
A block graph is a graph whose blocks are all cliques.
Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.
If thiscycle is an isometric cycle, we see that δ∗(G) > 0. If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Graphs = 0-Hyperbolic Graphs
Howorka (1979), Bandelt, Mulder (1986), Dress, Moulton, Steel(1997)
A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.
A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.
If thiscycle is an isometric cycle, we see that δ∗(G) > 0. If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Graphs = 0-Hyperbolic Graphs
Howorka (1979), Bandelt, Mulder (1986), Dress, Moulton, Steel(1997)
A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.
If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.
If thiscycle is an isometric cycle, we see that δ∗(G) > 0. If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Graphs = 0-Hyperbolic Graphs
Howorka (1979), Bandelt, Mulder (1986), Dress, Moulton, Steel(1997)
A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.
If thiscycle is an isometric cycle, we see that δ∗(G) > 0. If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Graphs = 0-Hyperbolic Graphs
Howorka (1979), Bandelt, Mulder (1986), Dress, Moulton, Steel(1997)
A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.
If thiscycle is an isometric cycle, we see that δ∗(G) > 0. If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Graphs = 0-Hyperbolic Graphs
Howorka (1979), Bandelt, Mulder (1986), Dress, Moulton, Steel(1997)
A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.If thiscycle is an isometric cycle, we see that δ∗(G) > 0.
If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Block Graphs = 0-Hyperbolic Graphs
Howorka (1979), Bandelt, Mulder (1986), Dress, Moulton, Steel(1997)
A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.If thiscycle is an isometric cycle, we see that δ∗(G) > 0. If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
12 -Hyperbolic Graphs
Bandelt and Chepoi (2003): A graph is 12 -hyperbolic if and
only if it does not contain a set of six special graphs asisometric subgraphs and all balls of the graph are convex.
Bandelt and Chepoi (2008): “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
12 -Hyperbolic Graphs
Bandelt and Chepoi (2003): A graph is 12 -hyperbolic if and
only if it does not contain a set of six special graphs asisometric subgraphs and all balls of the graph are convex.Bandelt and Chepoi (2008): “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Structure Tree Decomposition
The block tree is a decomposition of 1-connected graphsinto 2-connected blocks.
Tutte defined a tree decomposition of 2-connected graphsinto 3-connected blocks.Dunwoody and Krön (2010) define a structure tree thatcontains information about k -connectivity of a graph forany k .Can we relate the behavior of the hyperbolicity parameterto the structure tree decomposition?
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Structure Tree Decomposition
The block tree is a decomposition of 1-connected graphsinto 2-connected blocks.Tutte defined a tree decomposition of 2-connected graphsinto 3-connected blocks.
Dunwoody and Krön (2010) define a structure tree thatcontains information about k -connectivity of a graph forany k .Can we relate the behavior of the hyperbolicity parameterto the structure tree decomposition?
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Structure Tree Decomposition
The block tree is a decomposition of 1-connected graphsinto 2-connected blocks.Tutte defined a tree decomposition of 2-connected graphsinto 3-connected blocks.Dunwoody and Krön (2010) define a structure tree thatcontains information about k -connectivity of a graph forany k .
Can we relate the behavior of the hyperbolicity parameterto the structure tree decomposition?
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
0-Hyperbolic Graphs and Beyond
Structure Tree Decomposition
The block tree is a decomposition of 1-connected graphsinto 2-connected blocks.Tutte defined a tree decomposition of 2-connected graphsinto 3-connected blocks.Dunwoody and Krön (2010) define a structure tree thatcontains information about k -connectivity of a graph forany k .Can we relate the behavior of the hyperbolicity parameterto the structure tree decomposition?
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
Eccentricity
For any u ∈ V (G),
ecc(u) = maxv∈V (G)
uv ,
rad(G) = minu∈V (G)
ecc(u),
diam(G) = maxu∈V (G)
ecc(u).
The center of G, denoted C(G), is
{u ∈ V (G) : ecc(u) = rad(G)}.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
Eccentricity
For any u ∈ V (G),
ecc(u) = maxv∈V (G)
uv ,
rad(G) = minu∈V (G)
ecc(u),
diam(G) = maxu∈V (G)
ecc(u).
The center of G, denoted C(G), is
{u ∈ V (G) : ecc(u) = rad(G)}.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
Center
Camille Jordan (1869): Let G be a tree. Then xy ≤ 1 forany x , y ∈ C(G).
The center of a 12 -hyperbolic graph is convex.
Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): LetG be a graph. Then xy ≤ 4δ∗(G) + 1 for any x , y ∈ C(G).
Proof.For any x , y ∈ C(G), take vertices v and u such thatmax{xv , yv} ≤ d xy
2 e and uv = ecc(v).Note that uv ≥ rad(G) ≥ max{xu, yu}.It follows thatxy ≤ max{xu + yv − uv , xv + yu − uv}+ 2δ∗ ≤max{xv , yv}+ 2δ∗(G) ≤ dxy
2 e+ 2δ∗(G), implying thatxy ≤ 4δ∗(G) + 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
Center
Camille Jordan (1869): Let G be a tree. Then xy ≤ 1 forany x , y ∈ C(G).The center of a 1
2 -hyperbolic graph is convex.
Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): LetG be a graph. Then xy ≤ 4δ∗(G) + 1 for any x , y ∈ C(G).
Proof.For any x , y ∈ C(G), take vertices v and u such thatmax{xv , yv} ≤ d xy
2 e and uv = ecc(v).Note that uv ≥ rad(G) ≥ max{xu, yu}.It follows thatxy ≤ max{xu + yv − uv , xv + yu − uv}+ 2δ∗ ≤max{xv , yv}+ 2δ∗(G) ≤ dxy
2 e+ 2δ∗(G), implying thatxy ≤ 4δ∗(G) + 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
Center
Camille Jordan (1869): Let G be a tree. Then xy ≤ 1 forany x , y ∈ C(G).The center of a 1
2 -hyperbolic graph is convex.Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): LetG be a graph. Then xy ≤ 4δ∗(G) + 1 for any x , y ∈ C(G).
Proof.For any x , y ∈ C(G), take vertices v and u such thatmax{xv , yv} ≤ d xy
2 e and uv = ecc(v).Note that uv ≥ rad(G) ≥ max{xu, yu}.It follows thatxy ≤ max{xu + yv − uv , xv + yu − uv}+ 2δ∗ ≤max{xv , yv}+ 2δ∗(G) ≤ dxy
2 e+ 2δ∗(G), implying thatxy ≤ 4δ∗(G) + 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
Center
Camille Jordan (1869): Let G be a tree. Then xy ≤ 1 forany x , y ∈ C(G).The center of a 1
2 -hyperbolic graph is convex.Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): LetG be a graph. Then xy ≤ 4δ∗(G) + 1 for any x , y ∈ C(G).
Proof.For any x , y ∈ C(G), take vertices v and u such thatmax{xv , yv} ≤ d xy
2 e and uv = ecc(v).Note that uv ≥ rad(G) ≥ max{xu, yu}.It follows thatxy ≤ max{xu + yv − uv , xv + yu − uv}+ 2δ∗ ≤max{xv , yv}+ 2δ∗(G) ≤ d xy
2 e+ 2δ∗(G), implying thatxy ≤ 4δ∗(G) + 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
An Example from Gerard Chang (1991)
c s sc c csx sys
us
v
Figure: A chordal graph G whose center is marked with bullets.
(xv + yu, xu + yv , xy + uv) = (5, 4, 1), δ∗(G) = δ(x , y , u, v) = 12 .
The diameter of C(G) is 3 = 4× 12 + 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
Hyperbolicity and Small World
The hyperbolicity of a graph with diameter D is at mostbD
2 c.Proof
Many experiments say that some large practicalcommunication networks have surprisingly lowhyperbolicity (and so simple enough to deserve somemathematical study?).
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Eccentricity
Hyperbolicity and Small World
The hyperbolicity of a graph with diameter D is at mostbD
2 c.Proof
Many experiments say that some large practicalcommunication networks have surprisingly lowhyperbolicity (and so simple enough to deserve somemathematical study?).
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Cartesian Product
WU, Zhang (ܤ+) 2009: Let G1 and G2 be two graphs ofhyperbolicity 0. Then δ∗(G1�G2) = min(D1, D2), where D1 andD2 are the diameters of G1 and G2, respectively.
ry
ru
rvrxrr rr
r rrrδ∗ = δ(x , y , u, v) =
10− 62
= 2
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Breadth Property
For any two vertices x and y of a graph G, set brG(x , y) tobe max{uv : xu = xv = xy − uy = xy − vy}. The breadthof G, denoted br(G), is maxx ,y∈V (G) brG(x , y).
It is clear that br(G) = 0 when G is a tree.In general, given xu = xv = xy − yu = xy − yv , we havexu + yv = xv + yu = xy = (xy + uv)− uv .This says that br(G) ≤ 2δ∗(G).
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Breadth Property
For any two vertices x and y of a graph G, set brG(x , y) tobe max{uv : xu = xv = xy − uy = xy − vy}. The breadthof G, denoted br(G), is maxx ,y∈V (G) brG(x , y).It is clear that br(G) = 0 when G is a tree.
In general, given xu = xv = xy − yu = xy − yv , we havexu + yv = xv + yu = xy = (xy + uv)− uv .This says that br(G) ≤ 2δ∗(G).
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Breadth Property
For any two vertices x and y of a graph G, set brG(x , y) tobe max{uv : xu = xv = xy − uy = xy − vy}. The breadthof G, denoted br(G), is maxx ,y∈V (G) brG(x , y).It is clear that br(G) = 0 when G is a tree.In general, given xu = xv = xy − yu = xy − yv , we havexu + yv = xv + yu = xy = (xy + uv)− uv .
This says that br(G) ≤ 2δ∗(G).
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Breadth Property
For any two vertices x and y of a graph G, set brG(x , y) tobe max{uv : xu = xv = xy − uy = xy − vy}. The breadthof G, denoted br(G), is maxx ,y∈V (G) brG(x , y).It is clear that br(G) = 0 when G is a tree.In general, given xu = xv = xy − yu = xy − yv , we havexu + yv = xv + yu = xy = (xy + uv)− uv .This says that br(G) ≤ 2δ∗(G).
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Length of the Longest Chordless Cycle
A graph is k -chordal if it does not contain any induced n-cyclefor any n > k . The chordality of G, denoted lc(G), is theminimum integer k ≥ 2 such that G is k -chordal.
A 3-chordal graph is usually just called a chordal graph.An especially rich theory on chordal graphs has beendeveloped in an astonishingly wide area of mathematics andstatistics and other applied fields.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Length of the Longest Chordless Cycle
A graph is k -chordal if it does not contain any induced n-cyclefor any n > k . The chordality of G, denoted lc(G), is theminimum integer k ≥ 2 such that G is k -chordal.
A 3-chordal graph is usually just called a chordal graph.
An especially rich theory on chordal graphs has beendeveloped in an astonishingly wide area of mathematics andstatistics and other applied fields.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Length of the Longest Chordless Cycle
A graph is k -chordal if it does not contain any induced n-cyclefor any n > k . The chordality of G, denoted lc(G), is theminimum integer k ≥ 2 such that G is k -chordal.
A 3-chordal graph is usually just called a chordal graph.An especially rich theory on chordal graphs has beendeveloped in an astonishingly wide area of mathematics andstatistics and other applied fields.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Low Hyperbolicity ; Low Chordality
Indeed, take any graph G and form the new graph G′ by addingan additional vertex and connecting this new vertex with everyvertex of G. It is obvious that δ∗(G′) ≤ 1 while lc(G′) = lc(G) ifG is not a tree. Moreover, it is equally easy to see that G′ iseven 1
2 -hyperbolic if G does not have any induced 4-cycle(Koolen, Moulton, 2002)
Surely, this example does not preclude the possibility that formany important graph classes we can bound their chordalityfrom above in terms of their hyperbolicity.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Low Hyperbolicity ; Low Chordality
Indeed, take any graph G and form the new graph G′ by addingan additional vertex and connecting this new vertex with everyvertex of G. It is obvious that δ∗(G′) ≤ 1 while lc(G′) = lc(G) ifG is not a tree. Moreover, it is equally easy to see that G′ iseven 1
2 -hyperbolic if G does not have any induced 4-cycle(Koolen, Moulton, 2002)
Surely, this example does not preclude the possibility that formany important graph classes we can bound their chordalityfrom above in terms of their hyperbolicity.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
BKM Theorem
Theorem 1 (Brinkmann, Koolen, Moulton, 2001)
Every chordal graph is 1-hyperbolic and it is 12 -hyperbolic if and
only if it contains neither H1 nor H2 as an isometric subgraph.
ry
ru r v
rxra r b
rc r d
H1
ry
ru rvrxra rb
rc rdH2
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Low Chordality ⇒ Low Hyperbolicity
Theorem 2 (WU, Zhang, 2009)
For each k ≥ 4, all k-chordal graphs are b k2 c2 -hyperbolic.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tightness of Theorem 2
It is clear that if the bound claimed by Theorem 2 is tight fork = 4t (k = 4t − 2) then it is tight for k = 4t + 1 (k = 4t − 1).Consequently, Examples 3 and 4 to be presented below indeedmean that the bound reported in Theorem 2 is tight for everyk ≥ 4.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Example 3
The chordality of the k -cycle is surely k . Also recall that
δ∗(Ck ) =
b k
2 c2 , if k ≡ 0 (mod 4);
b k2 c2 + 1
2 , else.
Example 4
For any t ≥ 2 we set Ft to be the outerplanar graph obtainedfrom the 4t-cycle [v1v2 · · · v4t ] by adding the two edges {v1, v3}and {v2t+1, v2t+3}. Clearly, δ(v2, vt+2, v2t+2, v3t+2) = t − 1
2 .Furthermore, we can check that lc(Ft) = 4t − 2 and
δ∗(Ft) = t − 12 = δ(v2, vt+2, v2t+2, v3t+2) = lc(Ft )
4 =b lc(Ft )
2 c2 .
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
xy + uv = 3 + 4 = 7, xu + yv = xv + yu = 2 + 2 = 4 ⇒δ(x , y , u, v) = 3
2 .
rxra
ru
rc
rb rvrd
ry
Figure: The graph F2 has hyperbolicity 32 , tree-length 2 and
chordality 6.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Let C4, H1, H2, H3, H4 and H5 be the graphs displayed in nexttwo slides. It is simple to check that each of them hashyperbolicity 1 and is 5-chordal.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
ryru
rvrx
C4
ry
r
u rvrxra rb
rc rdH1
ry
ru rvrxra rb
rc rdH2
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
ry
ru rvrxrc rd
H3
ry
ru rvrxra rb
rc rdH4
ry
rc
ra rbrdru rv
rx
H5
Figure: Six 5-chordal graphs with hyperbolicity 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Theorem 5 (WU, Zhang, 2009)A 5-chordal graph has hyperbolicity one if and only if one ofC4, H1, H2, H3, H4, H5 appears as an isometric subgraph of it.
Together with Theorem 2, this generalizes the BKM Theorem(Theorem 1).
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
rr
r
r r
r rrr
G1
rr r
r r
r rr
rG2
rr
rr
rr
rG3
rr r
rr r
C6
Figure: Four graphs with hyperbolicity 1 and chordality 6.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Conjecture 6 (WU, Zhang, 2009)
A 6-chordal graph is 12 -hyperbolic if and only if it does not
contain any of a list of ten special graphsG1, G2, G3, C6, C4, Hi , i = 1, . . . , 5, as an isometric subgraph.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length
The tree-length of a graph G, denoted tl(G), wasintroduced by Dourisboure and Gavoille in 2007 and is theminimum integer k such that there is a chordal graph G′
satisfying V (G) = V (G′), E(G) ⊆ E(G′) andmax(dG(u, v) : dG′(u, v) = 1) = k . We use the conventionthat the tree-length of the graph with one vertex is 1.
It is straightforward from the definition that chordal graphsare exactly the graphs of tree-length 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length
The tree-length of a graph G, denoted tl(G), wasintroduced by Dourisboure and Gavoille in 2007 and is theminimum integer k such that there is a chordal graph G′
satisfying V (G) = V (G′), E(G) ⊆ E(G′) andmax(dG(u, v) : dG′(u, v) = 1) = k . We use the conventionthat the tree-length of the graph with one vertex is 1.It is straightforward from the definition that chordal graphsare exactly the graphs of tree-length 1.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree Decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 and has since beenextensively studied in both mathematics and lots of appliedfields. It specify a very nice way to view a graph as a tree.
A tree decomposition (T , S) of a graph G is a tree T such thateach vertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
(Vertex Covering) ∪v∈V (T )Sv = V (G).(Edge Covering) For any edge {u, w} ∈ E(G) there existsv ∈ V (T ) such that u, w ∈ Sv .
(Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree Decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 and has since beenextensively studied in both mathematics and lots of appliedfields. It specify a very nice way to view a graph as a tree.A tree decomposition (T , S) of a graph G is a tree T such thateach vertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
(Vertex Covering) ∪v∈V (T )Sv = V (G).(Edge Covering) For any edge {u, w} ∈ E(G) there existsv ∈ V (T ) such that u, w ∈ Sv .
(Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree Decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 and has since beenextensively studied in both mathematics and lots of appliedfields. It specify a very nice way to view a graph as a tree.A tree decomposition (T , S) of a graph G is a tree T such thateach vertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
(Vertex Covering) ∪v∈V (T )Sv = V (G).
(Edge Covering) For any edge {u, w} ∈ E(G) there existsv ∈ V (T ) such that u, w ∈ Sv .
(Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree Decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 and has since beenextensively studied in both mathematics and lots of appliedfields. It specify a very nice way to view a graph as a tree.A tree decomposition (T , S) of a graph G is a tree T such thateach vertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
(Vertex Covering) ∪v∈V (T )Sv = V (G).(Edge Covering) For any edge {u, w} ∈ E(G) there existsv ∈ V (T ) such that u, w ∈ Sv .
(Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree Decomposition
The concept of tree decompositions was introduced byRobertson and Seymour in 1984 and has since beenextensively studied in both mathematics and lots of appliedfields. It specify a very nice way to view a graph as a tree.A tree decomposition (T , S) of a graph G is a tree T such thateach vertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:
(Vertex Covering) ∪v∈V (T )Sv = V (G).(Edge Covering) For any edge {u, w} ∈ E(G) there existsv ∈ V (T ) such that u, w ∈ Sv .
(Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length is a Tree-likeness Parameter
The length of a tree decomposition of a graph G is themaximum distance in G between two vertices in the same bagof the decomposition. The length is a measure of the likenessof G to the tree T according to the given tree decomposition(T , S).
The tree-length of a graph G, tl(G), turns out to be the shortestlength of all tree decompositions of G.
Example 7 (Dourisboure and Gavoille (2007))
The tree-length of an n-cycle is dn3e.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length is a Tree-likeness Parameter
The length of a tree decomposition of a graph G is themaximum distance in G between two vertices in the same bagof the decomposition. The length is a measure of the likenessof G to the tree T according to the given tree decomposition(T , S).
The tree-length of a graph G, tl(G), turns out to be the shortestlength of all tree decompositions of G.
Example 7 (Dourisboure and Gavoille (2007))
The tree-length of an n-cycle is dn3e.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length is a Tree-likeness Parameter
The length of a tree decomposition of a graph G is themaximum distance in G between two vertices in the same bagof the decomposition. The length is a measure of the likenessof G to the tree T according to the given tree decomposition(T , S).
The tree-length of a graph G, tl(G), turns out to be the shortestlength of all tree decompositions of G.
Example 7 (Dourisboure and Gavoille (2007))
The tree-length of an n-cycle is dn3e.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Finite Characterization: Tree-length vs Hyperbolicity
From the tree decomposition definition of tree-length, we caneasily see that tl(H) ≤ tl(G) if H is a minor of G. By the GraphMinors Theorem of Robertson and Seymour, for any given k ,there exists a finite excluded minor characterization for thosegraphs whose tree length is at most k .
Hyperbolicity does not have the minor-monotone property.Instead, for every isometric subgraph H of a given graph G, wehave δ∗(H) ≤ δ∗(G). But, even 0-hyperbolic graphs may nothave a finite excluded isometric subgraph characterization.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Finite Characterization: Tree-length vs Hyperbolicity
From the tree decomposition definition of tree-length, we caneasily see that tl(H) ≤ tl(G) if H is a minor of G. By the GraphMinors Theorem of Robertson and Seymour, for any given k ,there exists a finite excluded minor characterization for thosegraphs whose tree length is at most k .
Hyperbolicity does not have the minor-monotone property.Instead, for every isometric subgraph H of a given graph G, wehave δ∗(H) ≤ δ∗(G). But, even 0-hyperbolic graphs may nothave a finite excluded isometric subgraph characterization.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length and Hyperbolicity are Comparable
The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés (2008).
Theorem 8The inequality tl(G) ≤ 12k + 8k log2 n + 17 holds for anyk-hyperbolic graph G with n vertices.
Theorem 9A graph G is k-hyperbolic provided its tree-length is no greaterthan k .
Proof of Theorem 9
Since chordal graphs have tree-length 1, the first part of theBKM Theorem (Theorem 1) directly follows from Theorem 9.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length and Hyperbolicity are Comparable
The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés (2008).
Theorem 8The inequality tl(G) ≤ 12k + 8k log2 n + 17 holds for anyk-hyperbolic graph G with n vertices.
Theorem 9A graph G is k-hyperbolic provided its tree-length is no greaterthan k .
Proof of Theorem 9
Since chordal graphs have tree-length 1, the first part of theBKM Theorem (Theorem 1) directly follows from Theorem 9.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length and Hyperbolicity are Comparable
The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés (2008).
Theorem 8The inequality tl(G) ≤ 12k + 8k log2 n + 17 holds for anyk-hyperbolic graph G with n vertices.
Theorem 9A graph G is k-hyperbolic provided its tree-length is no greaterthan k .
Proof of Theorem 9
Since chordal graphs have tree-length 1, the first part of theBKM Theorem (Theorem 1) directly follows from Theorem 9.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tree-length and Hyperbolicity are Comparable
The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés (2008).
Theorem 8The inequality tl(G) ≤ 12k + 8k log2 n + 17 holds for anyk-hyperbolic graph G with n vertices.
Theorem 9A graph G is k-hyperbolic provided its tree-length is no greaterthan k .
Proof of Theorem 9
Since chordal graphs have tree-length 1, the first part of theBKM Theorem (Theorem 1) directly follows from Theorem 9.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tightness of Theorem 9: δ∗(G2k+1,2k+1) = tl(G2k+1,2k+1)
Example 10
For any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1.
Dourisboure and Gavoile (2007)showed that the tree-length of Gn,m is min(n, m) if n 6= m orn = m is even and is n − 1 if n = m is odd. Our result onCartesian product shows that δ∗(Gm,n) = min(m, n)− 1. Thissays that Theorem 9 is tight.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tightness of Theorem 9: δ∗(G2k+1,2k+1) = tl(G2k+1,2k+1)
Example 10
For any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile (2007)showed that the tree-length of Gn,m is min(n, m) if n 6= m orn = m is even and is n − 1 if n = m is odd.
Our result onCartesian product shows that δ∗(Gm,n) = min(m, n)− 1. Thissays that Theorem 9 is tight.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tightness of Theorem 9: δ∗(G2k+1,2k+1) = tl(G2k+1,2k+1)
Example 10
For any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile (2007)showed that the tree-length of Gn,m is min(n, m) if n 6= m orn = m is even and is n − 1 if n = m is odd. Our result onCartesian product shows that δ∗(Gm,n) = min(m, n)− 1.
Thissays that Theorem 9 is tight.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Tightness of Theorem 9: δ∗(G2k+1,2k+1) = tl(G2k+1,2k+1)
Example 10
For any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile (2007)showed that the tree-length of Gn,m is min(n, m) if n 6= m orn = m is even and is n − 1 if n = m is odd. Our result onCartesian product shows that δ∗(Gm,n) = min(m, n)− 1. Thissays that Theorem 9 is tight.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Chordality and Tree-length
What follows is a result of Gavoille, Katz, Katz, Paul, Peleg(2001).
Theorem 11
If G is a k-chordal graph, then tl(G) ≤ bk2c.
Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique (Parra, Scheffler,1997). It is easy to check that each such new edge connectstwo points of distance at most b k
2c apart in G.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Chordality and Tree-length
What follows is a result of Gavoille, Katz, Katz, Paul, Peleg(2001).
Theorem 11
If G is a k-chordal graph, then tl(G) ≤ b k2c.
Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique (Parra, Scheffler,1997). It is easy to check that each such new edge connectstwo points of distance at most b k
2c apart in G.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Chordality and Tree-length
What follows is a result of Gavoille, Katz, Katz, Paul, Peleg(2001).
Theorem 11
If G is a k-chordal graph, then tl(G) ≤ b k2c.
Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique (Parra, Scheffler,1997).
It is easy to check that each such new edge connectstwo points of distance at most b k
2c apart in G.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Chordality and Tree-length
What follows is a result of Gavoille, Katz, Katz, Paul, Peleg(2001).
Theorem 11
If G is a k-chordal graph, then tl(G) ≤ b k2c.
Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique (Parra, Scheffler,1997). It is easy to check that each such new edge connectstwo points of distance at most b k
2c apart in G.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
Proof of Theorem 11
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u
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v
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C B
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|C|+ |B| = 4 + 6 ≤ k ⇒ uv ≤ |C| = 4 ≤ bk2c
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
An Easy proof of a Weaker Result
The following result is weaker than our Theorem 2. It wasnotified to us by Dragan and is presumably in the folklore.
Theorem 12
Every k-chordal graph is b k2c-hyperbolic.
Proof.
By Theorem 11, lc(G) ≤ k ⇒ tl(G) ≤ bk2c;
By Theorem 9, tl(G) ≤ bk2c ⇒ δ∗(G) ≤ bk
2c.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
An Easy proof of a Weaker Result
The following result is weaker than our Theorem 2. It wasnotified to us by Dragan and is presumably in the folklore.
Theorem 12
Every k-chordal graph is b k2c-hyperbolic.
Proof.
By Theorem 11, lc(G) ≤ k ⇒ tl(G) ≤ bk2c;
By Theorem 9, tl(G) ≤ bk2c ⇒ δ∗(G) ≤ bk
2c.
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
A Chordal Graph Sandwich Problem
Dourisboure and Gavoille (2007) posed the open problemthat whether or not
tl(G) ≤ d lc(G)
3e (1)
is true. They already knew that for any outerplanar graphG, tl(G) = d lc(G)
3 e holds.
The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .
Problem reformulation: For any graph G, is there always achordal graph H such that V (H) = V (G) = V (Gd lc(G)
3 e) andE(G) ⊆ E(H) ⊆ Gd lc(G)
3 e?
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
A Chordal Graph Sandwich Problem
Dourisboure and Gavoille (2007) posed the open problemthat whether or not
tl(G) ≤ d lc(G)
3e (1)
is true. They already knew that for any outerplanar graphG, tl(G) = d lc(G)
3 e holds.
The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .
Problem reformulation: For any graph G, is there always achordal graph H such that V (H) = V (G) = V (Gd lc(G)
3 e) andE(G) ⊆ E(H) ⊆ Gd lc(G)
3 e?
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
A Chordal Graph Sandwich Problem
Dourisboure and Gavoille (2007) posed the open problemthat whether or not
tl(G) ≤ d lc(G)
3e (1)
is true. They already knew that for any outerplanar graphG, tl(G) = d lc(G)
3 e holds.
The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .
Problem reformulation: For any graph G, is there always achordal graph H such that V (H) = V (G) = V (Gd lc(G)
3 e) andE(G) ⊆ E(H) ⊆ Gd lc(G)
3 e?
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
A Chordal Graph Sandwich Problem
Dourisboure and Gavoille (2007) posed the open problemthat whether or not
tl(G) ≤ d lc(G)
3e (1)
is true. They already knew that for any outerplanar graphG, tl(G) = d lc(G)
3 e holds.
The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .
Problem reformulation: For any graph G, is there always achordal graph H such that V (H) = V (G) = V (Gd lc(G)
3 e) andE(G) ⊆ E(H) ⊆ Gd lc(G)
3 e?
Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length
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Appendix
Proofδ(x , y , u, v) ≤ bD
2 c
Assume xy + uv ≥ xu + yv ≥ xv + yuand hence δ = (xy+uv)−(xu+yv)
2 .
By the triangle inequality,
xu + yu ≥ xy , ux + vx ≥ uv ,
xv + yv ≥ xy , vy + uy ≥ uv .
Summing up the above yields
(xu + yv) + (xv + yu) ≥ xy + uv .
This gives δ ≤ xy+uv4 ≤ D
2 .
When D is odd, we show thatδ = D
2 is impossible.
Otherwise, we havexv + yu = D, xu + yu = xy =D, ux + vx = uv = D, whichsays that 3D is an evennumber, a contradiction. Return
Appendix
Proof of Theorem 9
ry ∈ Y
ru ∈ U
rv ∈ V
rx ∈ X
rA rB
Let Q = {xu + yv , xv +yu, xy + uv}, α =xB + yB + uB + vB.
maxQ ≤ α + 2diam(B)
The largest two elementsof Q is at least α.
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