Hyperbolicity and Other Tree-likeness Parameters

Transcription

1 Hyperbolicity and Other Tree-likeness Parameters Yaokun Wu (Ljn) Department of Mathematics, Shanghai Jiao Tong University USTC Workshop on Graph Theory and Combinatorics May 30, 2010

2 Figure: Somewhere in FeiDong with those beautiful trees in our younger days...

3 Outline 1 Hyperbolicity Basic Definition Rooted versus Unrooted 0-Hyperbolic Graphs and Beyond 2 Easily Related Metric Graph Theory Concepts Eccentricity Cartesian Product Breadth Property 3 Chordality 4 Tree-length

4 Basic Definition Graph Metric Space We only consider simple, connected, unweighted, but not necessarily finite graphs. Any graph G together with the usual shortest-path metric on it, d G : V (G) V (G) {0, 1, 2,...}, gives rise to a metric space. We often use the shorthand xy for d G (x, y). Note that a pair of vertices x and y form an edge if and only if xy = 1.

5 Basic Definition A Function on Quartets For any vertices x, y, u, v of a graph G, put δ G (x, y, u, v) to be the difference between the largest and the second largest of the following three quantities xu + yv, 2 xv + yu, and 2 xy + uv. 2

6 Basic Definition δ(x, x, u, v) = 0 If x = y, then we have xu + yv= xv + yu= xu + xv uv =xy + uv.

7 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 provided x, y, u, v are not 4 different vertices.

8 Basic Definition Tree and the Four-Point Condition (4PC) x y a b u v xu + yv = xv + yu = xy + uv + 2ab δ(x, y, u, v) = 0

9 Basic Definition Hyperbolicity A graph G, viewed as a metric space, is δ-hyperbolic provided for any vertices x, y, u, v in G it holds δ(x, y, u, v) δ

10 Basic Definition Hyperbolicity A graph G, viewed as a metric space, is δ-hyperbolic provided for any vertices x, y, u, v in G it holds δ(x, y, u, v) δ

11 Basic Definition Hyperbolicity A graph G, viewed as a metric space, is δ-hyperbolic provided for any vertices x, y, u, v in G it holds δ(x, y, u, v) δ and the (Gromov) hyperbolicity of G, denoted δ (G), is the minimum half integer δ such that G is δ-hyperbolic.

12 Basic Definition Hyperbolicity A graph G, viewed as a metric space, is δ-hyperbolic provided for any vertices x, y, u, v in G it holds δ(x, y, u, v) δ and the (Gromov) hyperbolicity of G, denoted δ (G), is the minimum half integer δ such that G is δ-hyperbolic. Note that it may happen δ (G) =. But for a finite graph G, δ (G) is clearly finite and polynomial time computable.

13 Basic Definition Tree Metric Every tree has hyperbolicity zero.

14 Basic Definition Tree Metric Every tree has hyperbolicity zero. A metric space is a real tree (Tits 1977) if and only if it is path-connected and 0-hyperbolic.

15 Basic Definition Tree Metric Every tree has hyperbolicity zero. A metric space is a real tree (Tits 1977) if and only if it is path-connected and 0-hyperbolic. The hyperbolicity of a graph is a way to measure the additive distortion with which every four-points sub-metric of the given graph metric embeds into a tree metric, namely it is a tree-likeness parameter.

16 Basic Definition Weakly Geodesic Graphs A weakly geodesic graph is a graph in which any two vertices at distance two have exactly one common neighbor.

17 Basic Definition Weakly Geodesic Graphs A weakly geodesic graph is a graph in which any two vertices at distance two have exactly one common neighbor. If G is not weakly geodesic, we can find x, y, u, v such that xy = 2 and u, v are two different vertices lying in the geodesics between x and y.

18 Basic Definition Weakly Geodesic Graphs A weakly geodesic graph is a graph in which any two vertices at distance two have exactly one common neighbor. If G is not weakly geodesic, we can find x, y, u, v such that xy = 2 and u, v are two different vertices lying in the geodesics between x and y.this gives xy + uv 3 > 2 = xu + yv = xv + yu

19 Basic Definition Weakly Geodesic Graphs A weakly geodesic graph is a graph in which any two vertices at distance two have exactly one common neighbor. If G is not weakly geodesic, we can find x, y, u, v such that xy = 2 and u, v are two different vertices lying in the geodesics between x and y.this gives xy + uv 3 > 2 = xu + yv = xv + yu and hence δ (G) δ(x, y, u, v) 1 2.

20 Basic Definition Weakly Geodesic Graphs A weakly geodesic graph is a graph in which any two vertices at distance two have exactly one common neighbor. If G is not weakly geodesic, we can find x, y, u, v such that xy = 2 and u, v are two different vertices lying in the geodesics between x and y.this gives xy + uv 3 > 2 = xu + yv = xv + yu and hence δ (G) δ(x, y, u, v) 1 2. We now see that every 0-hyperbolic graph is weakly geodesic.

21 Basic Definition Cycle The hyperbolicity of the n-cycle C n is n to 1 modulo 4 and is n 4 else. if n is congruent

22 Basic Definition Cycle The hyperbolicity of the n-cycle C n is n to 1 modulo 4 and is n 4 else. if n is congruent In particular, C n has nonzero hyperbolicity for n > 3. Note that C n is weakly geodesic when n > 4.

23 Basic Definition Mikhail Gromov The concept of hyperbolicity comes from the work of Gromov in geometric group theory which encapsulates many of the global features of the geometry of complete, simply connected manifolds of negative curvature. M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, 1999.

24 Basic Definition Mikhail Gromov The concept of hyperbolicity comes from the work of Gromov in geometric group theory which encapsulates many of the global features of the geometry of complete, simply connected manifolds of negative curvature. M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, It is incredible what Mikhail Gromov can do, just with the triangle inequality. Dennis Sullivan

25 Rooted versus Unrooted Gromov Product For any vertex a V (G), the Gromov product, also known as the overlap function, of any two vertices x and y of G with respect to a is equal to 1 2 (xa + ya xy) and is denoted by (x y) a.

26 Rooted versus Unrooted Gromov Product For any vertex a V (G), the Gromov product, also known as the overlap function, of any two vertices x and y of G with respect 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 any x, y, v G it holds (x y) a min((x v) a, (y v) a ) δ.

27 Rooted versus Unrooted Gromov Product For any vertex a V (G), the Gromov product, also known as the overlap function, of any two vertices x and y of G with respect 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 any x, 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 is δ-hyperbolic with respect to every vertex of G.

28 Rooted versus Unrooted Farris Transform in Mathematics and Phylogenetics Think of a as the ancestor of all species, take a very large number ρ, then ρ (x y) a is an estimate for the number of years that have passed since the last common ancestor of x and y existed. 4 A. Dress, K.T. Huber, and V. Moulton x y lca(x,y) a Figure 2: The dotted lines depict D (a,ρ) (x,y) which can be thought of as an estimate for the number of years that have passed since lca(x,y) existed. The Farris transform based at a sends the given (graph) metric d G to the map in view of ρ+ 1 (DGC(x, y) DGC(x, a) DGC(y, a)) = ρ DGC(a, lca(x, y)) = 2 D ρ,a (d G ) : V (G) V (G) R : (x, y) ρ (x y) a. ρ D GC(a, v) D GC(lca(x, y), v) = ρ 0 D GC(lca(x, y), v) =

29 0-Hyperbolic Graphs and Beyond Block Tree A maximal connected subgraph without a cut vertex is called a block.

30 0-Hyperbolic Graphs and Beyond Block Tree A maximal connected subgraph without a cut vertex is called a block.the hyperbolicity of a graph is the maximum hyperbolicity of its blocks.

31 0-Hyperbolic Graphs and Beyond Block Tree A maximal connected subgraph without a cut vertex is called a block.the hyperbolicity of a graph is the maximum hyperbolicity of its blocks. e a b c d f a e c c d d b B 2 f B 1 B 3 B 1 c B 2 d B 3 g g The block tree of G has the cut vertices and blocks of G as its vertices and cb is an edge if c is a cut vertex inside the block B.

32 0-Hyperbolic Graphs and Beyond Block Tree A maximal connected subgraph without a cut vertex is called a block.the hyperbolicity of a graph is the maximum hyperbolicity of its blocks. e a b c d f a e c c d d b B 2 f B 1 B 3 B 1 c B 2 d B 3 g g The block tree of G has the cut vertices and blocks of G as its vertices and cb is an edge if c is a cut vertex inside the block B.The block tree is always a tree.

33 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.

34 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 block graphs.

35 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 block graphs. A graph is a block graph if and only if every cycle of the graph induces a clique.

36 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 block graphs. A graph is a block graph if and only if every cycle of the graph induces a clique. If a graph G is not a block graph, we can find a shortest cycle in it which does not induce a complete graph.

37 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 block graphs. A graph is a block graph if and only if every cycle of the graph induces a clique. If a graph G is not a block graph, we can find a shortest cycle in it which does not induce a complete graph.

38 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 block graphs. A graph is a block graph if and only if every cycle of the graph induces a clique. If a graph G is not a block graph, we can find a shortest cycle in it which does not induce a complete graph.if this cycle is an isometric cycle, we see that δ (G) > 0.

39 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 block graphs. A graph is a block graph if and only if every cycle of the graph induces a clique. If a graph G is not a block graph, we can find a shortest cycle in it which does not induce a complete graph.if this cycle is an isometric cycle, we see that δ (G) > 0. If this cycle is not an isometric cycle, we will find two maximal cliques of G sharing at least two vertices, which implies that G is not weakly geodesic and hence δ (G) > 0 also follows.

40 0-Hyperbolic Graphs and Beyond 1 2-Hyperbolic Graphs Bandelt and Chepoi (2003): A graph is 1 2 -hyperbolic if and only if it does not contain a set of six special graphs as isometric subgraphs and all balls of the graph are convex.

41 0-Hyperbolic Graphs and Beyond 1 2-Hyperbolic Graphs Bandelt and Chepoi (2003): A graph is 1 2-hyperbolic if and only if it does not contain a set of six special graphs as isometric subgraphs and all balls of the graph are convex. Bandelt and Chepoi (2008): a characterization of all 1-hyperbolic graphs by forbidden isometric subgraphs is not in sight, in as much as isometric cycles of lengths up to 7 may occur, thus complicating the picture.

42 0-Hyperbolic Graphs and Beyond Structure Tree Decomposition The block tree is a decomposition of 1-connected graphs into 2-connected blocks.

43 0-Hyperbolic Graphs and Beyond Structure Tree Decomposition The block tree is a decomposition of 1-connected graphs into 2-connected blocks. Tutte defined a tree decomposition of 2-connected graphs into 3-connected blocks.

44 0-Hyperbolic Graphs and Beyond Structure Tree Decomposition The block tree is a decomposition of 1-connected graphs into 2-connected blocks. Tutte defined a tree decomposition of 2-connected graphs into 3-connected blocks. Dunwoody and Krön (2010) define a structure tree that contains information about k-connectivity of a graph for any k.

45 0-Hyperbolic Graphs and Beyond Structure Tree Decomposition The block tree is a decomposition of 1-connected graphs into 2-connected blocks. Tutte defined a tree decomposition of 2-connected graphs into 3-connected blocks. Dunwoody and Krön (2010) define a structure tree that contains information about k-connectivity of a graph for any k. Can we relate the behavior of the hyperbolicity parameter to the structure tree decomposition?

46 Eccentricity Eccentricity For any u V (G), ecc(u) = max v V (G) uv, rad(g) = min ecc(u), u V (G) diam(g) = max u V (G) ecc(u).

47 Eccentricity Eccentricity For any u V (G), ecc(u) = max v V (G) uv, rad(g) = min ecc(u), u V (G) diam(g) = max u V (G) ecc(u). The center of G, denoted C(G), is {u V (G) : ecc(u) = rad(g)}.

48 Eccentricity Center Camille Jordan (1869): Let G be a tree. Then xy 1 for any x, y C(G).

49 Eccentricity Center Camille Jordan (1869): Let G be a tree. Then xy 1 for any x, y C(G). The center of a 1 2-hyperbolic graph is convex.

50 Eccentricity Center Camille Jordan (1869): Let G be a tree. Then xy 1 for any x, y C(G). The center of a 1 2-hyperbolic graph is convex. Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): Let G be a graph. Then xy 4δ (G) + 1 for any x, y C(G).

51 Eccentricity Center Proof. Camille Jordan (1869): Let G be a tree. Then xy 1 for any x, y C(G). The center of a 1 2-hyperbolic graph is convex. Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): Let G be a graph. Then xy 4δ (G) + 1 for any x, y C(G). For any x, y C(G), take vertices v and u such that max{xv, yv} xy 2 and uv = ecc(v). Note that uv rad(g) max{xu, yu}. It follows that xy max{xu + yv uv, xv + yu uv} + 2δ max{xv, yv} + 2δ (G) xy 2 + 2δ (G), implying that xy 4δ (G) + 1.

52 Eccentricity An Example from Gerard Chang (1991) x y u 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) = 1 2. The diameter of C(G) is 3 = v

53 Eccentricity Hyperbolicity and Small World The hyperbolicity of a graph with diameter D is at most D 2. Proof

54 Eccentricity Hyperbolicity and Small World The hyperbolicity of a graph with diameter D is at most D 2. Proof Many experiments say that some large practical communication networks have surprisingly low hyperbolicity (and so simple enough to deserve some mathematical study?).

55 Cartesian Product WU, Zhang (Ü +) 2009: Let G 1 and G 2 be two graphs of hyperbolicity 0. Then δ (G 1 G 2 ) = min(d 1, D 2 ), where D 1 and D 2 are the diameters of G 1 and G 2, respectively. x u v y δ = δ(x, y, u, v) = = 2

56 Breadth Property For any two vertices x and y of a graph G, set br G (x, y) to be max{uv : xu = xv = xy uy = xy vy}. The breadth of G, denoted br(g), is max x,y V (G) br G (x, y).

57 Breadth Property For any two vertices x and y of a graph G, set br G (x, y) to be max{uv : xu = xv = xy uy = xy vy}. The breadth of G, denoted br(g), is max x,y V (G) br G (x, y). It is clear that br(g) = 0 when G is a tree.

58 Breadth Property For any two vertices x and y of a graph G, set br G (x, y) to be max{uv : xu = xv = xy uy = xy vy}. The breadth of G, denoted br(g), is max x,y V (G) br G (x, y). It is clear that br(g) = 0 when G is a tree. In general, given xu = xv = xy yu = xy yv, we have xu + yv = xv + yu = xy = (xy + uv) uv.

59 Breadth Property For any two vertices x and y of a graph G, set br G (x, y) to be max{uv : xu = xv = xy uy = xy vy}. The breadth of G, denoted br(g), is max x,y V (G) br G (x, y). It is clear that br(g) = 0 when G is a tree. In general, given xu = xv = xy yu = xy yv, we have xu + yv = xv + yu = xy = (xy + uv) uv. This says that br(g) 2δ (G).

60 Length of the Longest Chordless Cycle A graph is k-chordal if it does not contain any induced n-cycle for any n > k. The chordality of G, denoted lc(g), is the minimum integer k 2 such that G is k-chordal.

61 Length of the Longest Chordless Cycle A graph is k-chordal if it does not contain any induced n-cycle for any n > k. The chordality of G, denoted lc(g), is the minimum integer k 2 such that G is k-chordal. A 3-chordal graph is usually just called a chordal graph.

62 Length of the Longest Chordless Cycle A graph is k-chordal if it does not contain any induced n-cycle for any n > k. The chordality of G, denoted lc(g), is the minimum 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 been developed in an astonishingly wide area of mathematics and statistics and other applied fields.

63 Low Hyperbolicity Low Chordality Indeed, take any graph G and form the new graph G by adding an additional vertex and connecting this new vertex with every vertex of G. It is obvious that δ (G ) 1 while lc(g ) = lc(g) if G is not a tree. Moreover, it is equally easy to see that G is even 1 2-hyperbolic if G does not have any induced 4-cycle (Koolen, Moulton, 2002)

64 Low Hyperbolicity Low Chordality Indeed, take any graph G and form the new graph G by adding an additional vertex and connecting this new vertex with every vertex of G. It is obvious that δ (G ) 1 while lc(g ) = lc(g) if G is not a tree. Moreover, it is equally easy to see that G is even 1 2-hyperbolic if G does not have any induced 4-cycle (Koolen, Moulton, 2002) Surely, this example does not preclude the possibility that for many important graph classes we can bound their chordality from above in terms of their hyperbolicity.

65 BKM Theorem Theorem 1 (Brinkmann, Koolen, Moulton, 2001) Every chordal graph is 1-hyperbolic and it is 1 2-hyperbolic if and only if it contains neither H 1 nor H 2 as an isometric subgraph. u a c x y b v d a u c x y b v d H 1 H 2

66 Low Chordality Low Hyperbolicity Theorem 2 (WU, Zhang, 2009) For each k 4, all k-chordal graphs are k 2 2 -hyperbolic.

67 Tightness of Theorem 2 It is clear that if the bound claimed by Theorem 2 is tight for k = 4t (k = 4t 2) then it is tight for k = 4t + 1 (k = 4t 1). Consequently, Examples 3 and 4 to be presented below indeed mean that the bound reported in Theorem 2 is tight for every k 4.

68 Example 3 The chordality of the k-cycle is surely k. Also recall that k δ 2 (C k ) = 2, if k 0 (mod 4); k , else. Example 4 For any t 2 we set F t to be the outerplanar graph obtained from the 4t-cycle [v 1 v 2 v 4t ] by adding the two edges {v 1, v 3 } and {v 2t+1, v 2t+3 }. Clearly, δ(v 2, v t+2, v 2t+2, v 3t+2 ) = t 1 2. Furthermore, we can check that lc(f t ) = 4t 2 and δ (F t ) = t 1 2 = δ(v 2, v t+2, v 2t+2, v 3t+2 ) = lc(f t ) 4 = lc(ft ) 2 2.

69 xy + uv = = 7, xu + yv = xv + yu = = 4 δ(x, y, u, v) = 3 2. x b v a d u c y Figure: The graph F 2 has hyperbolicity 3 2, tree-length 2 and chordality 6.

70 Let C 4, H 1, H 2, H 3, H 4 and H 5 be the graphs displayed in next two slides. It is simple to check that each of them has hyperbolicity 1 and is 5-chordal.

71 x u C 4 v y u a c x y H 1 a b u v c d x y H 2 b v d

72 u c x v d y H 3 x a b a u v u c d c y H 4 x b v d y H 5 Figure: Six 5-chordal graphs with hyperbolicity 1.

73 Theorem 5 (WU, Zhang, 2009) A 5-chordal graph has hyperbolicity one if and only if one of C 4, H 1, H 2, H 3, H 4, H 5 appears as an isometric subgraph of it. Together with Theorem 2, this generalizes the BKM Theorem (Theorem 1).

74 G 1 G 2 G 3 C 6 Figure: Four graphs with hyperbolicity 1 and chordality 6.

75 Conjecture 6 (WU, Zhang, 2009) A 6-chordal graph is 1 2 -hyperbolic if and only if it does not contain any of a list of ten special graphs G 1, G 2, G 3, C 6, C 4, H i, i = 1,..., 5, as an isometric subgraph.

76 Tree-length The tree-length of a graph G, denoted tl(g), was introduced by Dourisboure and Gavoille in 2007 and is the minimum integer k such that there is a chordal graph G satisfying V (G) = V (G ), E(G) E(G ) and max(d G (u, v) : d G (u, v) = 1) = k. We use the convention that the tree-length of the graph with one vertex is 1.

77 Tree-length The tree-length of a graph G, denoted tl(g), was introduced by Dourisboure and Gavoille in 2007 and is the minimum integer k such that there is a chordal graph G satisfying V (G) = V (G ), E(G) E(G ) and max(d G (u, v) : d G (u, v) = 1) = k. We use the convention that the tree-length of the graph with one vertex is 1. It is straightforward from the definition that chordal graphs are exactly the graphs of tree-length 1.

78 Tree Decomposition The concept of tree decompositions was introduced by Robertson and Seymour in 1984 and has since been extensively studied in both mathematics and lots of applied fields. It specify a very nice way to view a graph as a tree.

79 Tree Decomposition The concept of tree decompositions was introduced by Robertson and Seymour in 1984 and has since been extensively studied in both mathematics and lots of applied fields. 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 that each vertex v of T corresponds to a bag S v V (G) and the following conditions are satisfied:

80 Tree Decomposition The concept of tree decompositions was introduced by Robertson and Seymour in 1984 and has since been extensively studied in both mathematics and lots of applied fields. 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 that each vertex v of T corresponds to a bag S v V (G) and the following conditions are satisfied: (Vertex Covering) v V (T ) S v = V (G).

81 Tree Decomposition The concept of tree decompositions was introduced by Robertson and Seymour in 1984 and has since been extensively studied in both mathematics and lots of applied fields. 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 that each vertex v of T corresponds to a bag S v V (G) and the following conditions are satisfied: (Vertex Covering) v V (T ) S v = V (G). (Edge Covering) For any edge {u, w} E(G) there exists v V (T ) such that u, w S v.

82 Tree Decomposition The concept of tree decompositions was introduced by Robertson and Seymour in 1984 and has since been extensively studied in both mathematics and lots of applied fields. 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 that each vertex v of T corresponds to a bag S v V (G) and the following conditions are satisfied: (Vertex Covering) v V (T ) S v = V (G). (Edge Covering) For any edge {u, w} E(G) there exists v V (T ) such that u, w S v. (Running Intersection Property) For any u V (G), {v V (T ) : u S v } induces a subtree of T. In other words, for any v, w V (T ), S v S w can be seen in every bag along the path connecting v and w in T.

83 Tree-length is a Tree-likeness Parameter The length of a tree decomposition of a graph G is the maximum distance in G between two vertices in the same bag of the decomposition. The length is a measure of the likeness of G to the tree T according to the given tree decomposition (T, S).

84 Tree-length is a Tree-likeness Parameter The length of a tree decomposition of a graph G is the maximum distance in G between two vertices in the same bag of the decomposition. The length is a measure of the likeness of 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 shortest length of all tree decompositions of G.

85 Tree-length is a Tree-likeness Parameter The length of a tree decomposition of a graph G is the maximum distance in G between two vertices in the same bag of the decomposition. The length is a measure of the likeness of 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 shortest length of all tree decompositions of G. Example 7 (Dourisboure and Gavoille (2007)) The tree-length of an n-cycle is n 3.

86 Finite Characterization: Tree-length vs Hyperbolicity From the tree decomposition definition of tree-length, we can easily see that tl(h) tl(g) if H is a minor of G. By the Graph Minors Theorem of Robertson and Seymour, for any given k, there exists a finite excluded minor characterization for those graphs whose tree length is at most k.

87 Finite Characterization: Tree-length vs Hyperbolicity From the tree decomposition definition of tree-length, we can easily see that tl(h) tl(g) if H is a minor of G. By the Graph Minors Theorem of Robertson and Seymour, for any given k, there exists a finite excluded minor characterization for those graphs 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, we have δ (H) δ (G). But, even 0-hyperbolic graphs may not have a finite excluded isometric subgraph characterization.

88 Tree-length and Hyperbolicity are Comparable The following are two results from Chepoi, Dragan, Estellon, Habib, Vaxés (2008).

89 Tree-length and Hyperbolicity are Comparable The following are two results from Chepoi, Dragan, Estellon, Habib, Vaxés (2008). Theorem 8 The inequality tl(g) 12k + 8k log 2 n + 17 holds for any k-hyperbolic graph G with n vertices.

90 Tree-length and Hyperbolicity are Comparable The following are two results from Chepoi, Dragan, Estellon, Habib, Vaxés (2008). Theorem 8 The inequality tl(g) 12k + 8k log 2 n + 17 holds for any k-hyperbolic graph G with n vertices. Theorem 9 A graph G is k-hyperbolic provided its tree-length is no greater than k. Proof of Theorem 9

91 Tree-length and Hyperbolicity are Comparable The following are two results from Chepoi, Dragan, Estellon, Habib, Vaxés (2008). Theorem 8 The inequality tl(g) 12k + 8k log 2 n + 17 holds for any k-hyperbolic graph G with n vertices. Theorem 9 A graph G is k-hyperbolic provided its tree-length is no greater than k. Proof of Theorem 9 Since chordal graphs have tree-length 1, the first part of the BKM Theorem (Theorem 1) directly follows from Theorem 9.

92 Tightness of Theorem 9: δ (G 2k+1,2k+1 ) = tl(g 2k+1,2k+1 ) Example 10 For any two natural numbers m and n, the grid graph G m,n is the graph with vertex set {1, 2,..., m} {1, 2,..., n} and (i 1, i 2 ) and (j 1, j 2 ) are adjacent in G m,n if any only if (i 1 j 1 ) 2 + (j 1 j 2 ) 2 = 1.

93 Tightness of Theorem 9: δ (G 2k+1,2k+1 ) = tl(g 2k+1,2k+1 ) Example 10 For any two natural numbers m and n, the grid graph G m,n is the graph with vertex set {1, 2,..., m} {1, 2,..., n} and (i 1, i 2 ) and (j 1, j 2 ) are adjacent in G m,n if any only if (i 1 j 1 ) 2 + (j 1 j 2 ) 2 = 1. Dourisboure and Gavoile (2007) showed that the tree-length of G n,m is min(n, m) if n m or n = m is even and is n 1 if n = m is odd.

94 Tightness of Theorem 9: δ (G 2k+1,2k+1 ) = tl(g 2k+1,2k+1 ) Example 10 For any two natural numbers m and n, the grid graph G m,n is the graph with vertex set {1, 2,..., m} {1, 2,..., n} and (i 1, i 2 ) and (j 1, j 2 ) are adjacent in G m,n if any only if (i 1 j 1 ) 2 + (j 1 j 2 ) 2 = 1. Dourisboure and Gavoile (2007) showed that the tree-length of G n,m is min(n, m) if n m or n = m is even and is n 1 if n = m is odd. Our result on Cartesian product shows that δ (G m,n ) = min(m, n) 1.

95 Tightness of Theorem 9: δ (G 2k+1,2k+1 ) = tl(g 2k+1,2k+1 ) Example 10 For any two natural numbers m and n, the grid graph G m,n is the graph with vertex set {1, 2,..., m} {1, 2,..., n} and (i 1, i 2 ) and (j 1, j 2 ) are adjacent in G m,n if any only if (i 1 j 1 ) 2 + (j 1 j 2 ) 2 = 1. Dourisboure and Gavoile (2007) showed that the tree-length of G n,m is min(n, m) if n m or n = m is even and is n 1 if n = m is odd. Our result on Cartesian product shows that δ (G m,n ) = min(m, n) 1. This says that Theorem 9 is tight.

96 Chordality and Tree-length What follows is a result of Gavoille, Katz, Katz, Paul, Peleg (2001).

97 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) k 2.

98 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) k 2. Proof. To obtain a minimal triangulation of G, it suffices to select a maximal set of pairwise parallel minimal separators of G and add edges to make each of them a clique (Parra, Scheffler, 1997).

99 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) k 2. Proof. To obtain a minimal triangulation of G, it suffices to select a maximal set of pairwise parallel minimal separators of G and add edges to make each of them a clique (Parra, Scheffler, 1997). It is easy to check that each such new edge connects two points of distance at most k 2 apart in G.

100 Proof of Theorem 11 C u B v C + B = k uv C = 4 k 2

101 An Easy proof of a Weaker Result The following result is weaker than our Theorem 2. It was notified to us by Dragan and is presumably in the folklore. Theorem 12 Every k-chordal graph is k 2 -hyperbolic.

102 An Easy proof of a Weaker Result The following result is weaker than our Theorem 2. It was notified to us by Dragan and is presumably in the folklore. Theorem 12 Every k-chordal graph is k 2 -hyperbolic. Proof. By Theorem 11, lc(g) k tl(g) k 2 ; By Theorem 9, tl(g) k 2 δ (G) k 2.

103 A Chordal Graph Sandwich Problem Dourisboure and Gavoille (2007) posed the open problem that whether or not tl(g) lc(g) 3 (1) is true. They already knew that for any outerplanar graph G, tl(g) = lc(g) 3 holds.

104 A Chordal Graph Sandwich Problem Dourisboure and Gavoille (2007) posed the open problem that whether or not tl(g) lc(g) 3 (1) is true. They already knew that for any outerplanar graph G, tl(g) = lc(g) 3 holds. The kth-power of a graph G, denoted G k, is the graph with V (G) as vertex set and there is an edge connecting two vertices u and v if and only if d G (u, v) k.

105 A Chordal Graph Sandwich Problem Dourisboure and Gavoille (2007) posed the open problem that whether or not tl(g) lc(g) 3 (1) is true. They already knew that for any outerplanar graph G, tl(g) = lc(g) 3 holds. The kth-power of a graph G, denoted G k, is the graph with V (G) as vertex set and there is an edge connecting two vertices u and v if and only if d G (u, v) k.

106 A Chordal Graph Sandwich Problem Dourisboure and Gavoille (2007) posed the open problem that whether or not tl(g) lc(g) 3 (1) is true. They already knew that for any outerplanar graph G, tl(g) = lc(g) 3 holds. The kth-power of a graph G, denoted G k, is the graph with V (G) as vertex set and there is an edge connecting two vertices u and v if and only if d G (u, v) k. Problem reformulation: For any graph G, is there always a chordal graph H such that V (H) = V (G) = V (G lc(g) 3 ) and E(G) E(H) G lc(g) 3?

107 oè: 5qY6c6 ) àl6l ò õc± Xb1ðc ù 6cqY _

108 Appendix Proof δ(x, y, u, v) D 2 Assume xy + uv xu + yv xv + yu and 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+uv 4 D 2. When D is odd, we show that δ = D 2 is impossible. Otherwise, we have xv + yu = D, xu + yu = xy = D, ux + vx = uv = D, which says that 3D is an even number, a contradiction. Return

109 Appendix Proof of Theorem 9 Let Q = {xu + yv, xv + yu, xy + uv}, α = xb + yb + ub + vb. x X A y Y u U B v V max Q α + 2diam(B) The largest two elements of Q is at least α. Return