Chordality properties and hyperbolicity on graphs

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. In this paper we study the relation between the hyperbolicity of the graph and some chordality properties which are natural generalizations of being chordal. We find chordality properties that are weaker and stronger than being $\delta$-hyperbolic. Moreover, we obtain a characterization of being hyperbolic on terms of a chordality property on the triangles.

Given a metric space (X, d), a geodesic from x ∈ X to y ∈ X is an isometry, γ, from a closed interval [0, l] ⊂ R to X such that γ(0) = x, γ(l) = y. We will also call geodesic to the image of γ. X is a geodesic metric space if for every x, y ∈ X there exists a geodesic joining x and y; any of these geodesics will be denoted as [xy] although this notation is ambiguous since geodesics need not be unique.
By a metric graph we mean a graph G equipped with a length metric by considering every edge e ∈ E(G) as isometric to an interval [0, l e ]. Thus, the interior points of the edges are also considered points in G. Then, for any pair of points in x, y ∈ G, the distance d(x, y) will be the length of the shortest path in G joining x and y. In this paper we will assume that the graphs are connected and locally finite (i.e. each ball intersects a finite number of edges) and there is no restriction on the length of the edges. In this case, the metric graph is a geodesic metric space.
There are several definitions of Gromov δ-hyperbolic space which are equivalent although the constant δ may appear multiplied by some constant (see [10]). We are going to use the characterization of Gromov hyperbolicity for geodesic metric spaces given by the Rips condition on the geodesic triangles. If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X, the union of three geodesics [x 1 x 2 ], [x 2 x 3 ] and [x 3 x 1 ] is called a geodesic triangle and will be denoted by T = {x 1 , x 2 , x 3 }. T is δ-thin if any side of T is contained in the δneighborhood of the union of the two other sides. The space X is δ-hyperbolic if every geodesic triangle in X is δ-thin. We denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X) := inf{δ | every triangle in X is δ-thin}. We say that X is hyperbolic if X is δ-hyperbolic for some δ ≥ 0. A triangle with two identical vertices is called a "bigon".
A graph G is chordal if every induced cycle has at most three edges. In [7], the authors prove that chordal graphs are hyperbolic giving an upper bound for the hyperbolicity constant. In [40], Wu and Zhang extend this result for a generalized version of chordality. They prove that k-chordal graphs are hyperbolic where a graph is k-chordal if every induced cycle has at most k edges. In [1], the authors define the more general properties of being (k, m)-edge-chordal and (k, k 2 )-path-chordal and prove that every (k, m)edge-chordal graph is hyperbolic and that every hyperbolic graph is (k, k 2 )path-chordal. Herein, we continue this work and define being ε-densely (k, m)-path-chordal and ε-densely (k, m)-path-chordal obtaining that (see Theorems 2.3, 2.4 and 2.8) We also provide examples showing that all these implications are strict, this is, the converse is not true.
Moreover, we give a characterization of hyperbolicity in terms of a chordality property on the triangles: Theorem 3.2 states that a metric graph G is δ-hyperbolic if and only if G is ε-densely (k, m)-path-chordal on the triangles.
The properties and implications studied in this paper are summarized in Figure 1.
k-path-ch. 2 ε-densely path-chordal graphs Consider a metric graph (G, d). By a cycle in a graph we mean a simple closed curve, this is, a path defined by a sequence of vertices which are all different except for the first one and the last one which are the same. A shortcut in a cycle C is a path σ joining two vertices p, q in C such that L(σ) < d C (p, q) where d C denotes the length metric on C.
In this case, we say that p, q are shortcut vertices in C associated to σ.
Let us recall two definitions from [1]: Given two constants k, m ≥ 0, a metric graph G is (k, m)-edge-chordal if for any cycle C in G with length L(C) ≥ k there exists an edge-shortcut e with length L(e) ≤ m. The graph G is edge-chordal if there exist constants k, m ≥ 0 such that G is (k, m)-edge-chordal. Notice that for a graph with edges of length 1, being chordal is equivalent to being (4, 1)-edge-chordal and being k-chordal in the sense of [40] is equivalent to being (k + 1, 1)edge-chordal.
A metric graph G is (k, k 2 )-path-chordal if for any cycle C in G with L(C) ≥ k there exists a shortcut σ of C such that L(σ) ≤ k 2 . Notice that in [1] this is called simply k-path-chordal. However, new definitions are introduced herein and this property has been renamed. Given a metric space (X, d) and any ε > 0, a subset A ⊂ X is ε-dense if for every x ∈ X there exist some a ∈ A such that d(a, x) < ε.
for every cycle C with length L(C) ≥ k, there exist strict shortcuts σ 1 , ..., σ r with L(σ i ) ≤ m and such that their associated shortcut vertices define an ε-dense subset in (C, d C ).
Proof. Clearly, being (k, m)-edge-chordal implies being (k, m)-path-chordal. Thus, it suffices to check that there is some ε > 0 such that, on every cycle C with length at least k, the shortcut vertices associated to the edge-shortcuts with length at most m are ε-dense.
Let ε := k 2 and let p be any point in C. Since G is (k, m)-edge-chordal, there is an edge xy with L(xy) ≤ m which is a shortcut in C. If either x or y is in B C (p, k 2 ), we are done. Otherwise, let us define a cycle C 1 by joining the path in C from x to y containing p and the edge xy. Since x, y / ∈ B C (p, k 2 ), C 1 has length at least k and there is another edge, x y , with L(x y ) ≤ m which defines a shortcut in C 1 . Since xy is an edge, x , y ∈ C and x y is also an edge-shortcut in C. If either x or y is in B C (p, k 2 ), we are done. This process can be repeated and, since the graph is finite, we will finally obtain an edge-shortcut in C such that one of its vertices is contained in B C (p, k 2 ) finishing the proof.
The converse is not true. Let P 3 be the path graph with (adjacent) vertices v 1 , v 2 , v 3 , and G the Cartesian product graph G = Z2P 3 with L(e) = 1 for every e ∈ E(G). As it was shown in [1], G is not edge-chordal although it is 5 2 -hyperbolic and (5, 5 2 )-path chordal. Let us see that G is 3-densely (5, 2)-path-chordal. Let C be any cycle in G with L(C) ≥ 5 and let (z i , v j ) be any vertex in C. It suffices to check that there is a shortcut vertex in B C (z i , v j ), 5 2 associated to a shortcut with length at most 2.
is a shortcut with L(σ) = 1 and (z i , z j−1 ) is a shortcut vertex associated to σ and contained in B C (z i , v j ), 5 2 .
Proof. Suppose that G is ε-densely (k, m)-path-chordal. Consider any geodesic triangle T = {x, y, z} in G. If L(T ) < k, it follows that every side of the triangle has length at most k 2 . Therefore, the hyperbolic constant is at most The converse is not true. To build an example of a δ-hyperbolic graph which is not ε-densely (k, m)-path-chordal let us recall here the construction of the hyperbolic approximation of a metric space introduced by S. Buyalo and V. Schroeder in [10]. The hyperbolic approximation of a metric space is a special kind of hyperbolic cone, see [8], which is defined in general for non-necessarily bounded metric spaces.
A subset A in a metric space Z is called r-separated, r > 0, if d(a, a ) ≥ r for any distinct a, a ∈ A. Note that if A is maximal with this property, then the union ∪ a∈A B r (a) covers Z.
A hyperbolic approximation of a metric space Z is a graph X which is defined as follows. Fix a positive r ≤ 1 6 which is called the parameter of X. For every k ∈ Z, let A k ∈ Z be a maximal r k -separated set. For every a ∈ A k , consider the ball B(a, 2r k ) ⊂ Z. Let us fix the set V as the union, for k ∈ Z, of the set of balls B(a, 2r k ), a ∈ A k . Let us denote the corresponding ball simply by B(v).
Let V be the vertex set of the graph X. Vertices v, v are connected by an edge if and only if they either belong to the same level, V k , and the closed ballsB(v),B(v ) intersect,B(v)∩B(v ) = ∅, or they lie on neighboring levels V k , V k+1 and the ball of the upper level, V k+1 , is contained in the ball of the lower level, V k .
An edge vv ⊂ X is called horizontal if its vertices belong to the same level, v, v ∈ V k for some k ∈ Z. Other edges are called radial. Consider the path metric on X for which every edge has length 1. There is a natural level function l : Note that any (finite or infinite) sequence {v k } ∈ V such that v k v k+1 is a radial edge for every k and such that the level function l is monotone along {v k }, is the vertex sequence of a geodesic in X. Such a geodesic is called radial.
The following technical lemma is very useful to understand the geodesics in X.
Lemma 2.6. [10, Lemma 6.2.6] Any vertices v, v ∈ V can be connected in X by a geodesic which contains at most one horizontal edge. If there is such an edge, then it lies on the lowest level of the geodesic. Now, let us consider the following hyperbolic approximation of the Euclidean real line.
Let r = 1 6 and A k := {mr k : m ∈ Z}. By Proposition 2.5, the resultant hyperbolic approximation G is δ-hyperbolic. Let us see that G not ε-densely (k, m)-path-chordal.
Consider the cycle C n ∈ G defined as follows. First, consider the vertices v 0 , v 1 , ..., v 6 n in V 0 such that B(v i ) = B(i, 2) for every 0 ≤ i ≤ 6 n and the horizontal edges v i−1 v i for every 1 ≤ i ≤ 6 n . Also, for every 0 < k < n consider the vertices v k , v k in V −k such that B(v k ) = B(6 k , 2 · 6 k ) and B(v k ) = B(6 n − 6 k , 2 · 6 k ). Then, consider the radial edges v 0 v 1 , v 6 n v 1 and v k−1 v k , v k−1 v k for every 1 < k < n. Finally, to complete the cycle, consider the horizontal edge v n−1 v n−1 .
Let γ be the path in C n from v 0 to v 6 n given by the radial geodesic from v 0 to v n−1 , the horizontal edge v n−1 v n−1 and the radial geodesic from v n−1 to v 6 n . Let us prove that γ is a geodesic.
Notice that L(γ) = 2n − 1. Suppose that γ is a geodesic in X from v 0 to v 6 n with L(γ ) < 2n − 1. By Lemma 2.6 we may assume that γ contains at most one horizontal edge which lies in the lowest level, −m. Moreover, we may assume that γ begins with m radial edges from level 0 to level −m, then it may have one horizontal edge or not, and then it has m radial edges from level −m to level 0.
Let v ∈ C n such that l(v) = −n + 1 and let ε = n−1 2 . Then, for every w ∈ C n such that d Cn (v, w) < ε, l(w) ≤ − n−1 2 , and for every shortcut σ such that w is a shortcut vertex associated to σ, L(σ) ≥ n−1 2 . Thus, G is not ε-densely (k, m)-path-chordal for any constants ε, k, m. Definition 2.7. A metric graph (G, d) is ε-densely k-path-chordal if for every cycle C with length L(C) ≥ k, there exist strict shortcuts σ 1 , ..., σ r such that their associated shortcut vertices define an ε-dense subset in (C, d C ).
Let C be any cycle with L(C) = 4δ. Then, if there is a shortcut vertex in C, we are done. Suppose there are no shortcut vertices in C. Therefore, there exist two points x, y ∈ C such that d(x, y) = L(C) 2 = 2δ and the cycle C is formed by two different geodesics, γ 1 , γ 2 joining x, y. Consider the bigon given by γ 1 ∪ γ 2 . Since G is δ-hyperbolic, for every point p in γ 1 , d(p, γ 2 ) < δ. However, if m is the middle point in γ 1 , then it is clear that d C (m, γ 2 ) = δ. Therefore, there must be a shortcut σ joining γ 1 and γ 2 leading to contradiction. Now, let C be any cycle with 4δ < L(C). Let us suppose that there is a point x ∈ C such that there are no shortcut vertices in B C (x, 2δ). Let a, b be the points in C such that d C (a, x) = 2δ = d C (b, x). Since there are no shortcut vertices in B C (x, 2δ), the restriction of the cycle, C ∩ B C (x, 2δ), defines two geodesics [xa] and [xb] with length 2δ contained in C.
Also, since there are no shortcut vertices in B C (x, 2δ), for every point y in C\B C (x, 2δ), the geodesic [xy] either contains [xa] or [xb]. In fact, since C\B C (x, 2δ) is connected, there exist two sequences p n , q n with d C (p n , q n ) ≤ Remark 2.9. Being ε-densely k-path-chordal does not imply being (k, k 2 )path-chordal (this is, k-path-chordal as defined in [1]). Also, it does not imply being δ-hyperbolic.
Consider the Cartesian product graph G = N2N with L(e) = 1 for every edge e in G.
First, notice that G is not (k, k 2 )-path-chordal for any k ≥ 5. Consider any n ≥ k and let C n be the geodesic square defined by the vertices (1, 1), (n, 1), (n, n), (1, n). Thus, C n is a cycle with length L(C n ) = 4n − 4 > k. It is trivial to check that every shortcut σ in C n must join two opposite sides of the square and therefore, L(σ) ≥ n − 1 > k 2 . It is trivial to check that G is not δ-hyperbolic either. Now, let us see that G is 2-densely 6-path chordal. Let C be any cycle with L(C) ≥ 6. Let (p, q) be any vertex in C and let us prove that either (p, q) is a shortcut vertex or it is adjacent to a shortcut vertex. Case 1. Let us suppose that (p, q) and the two adjacent vertices in C are aligned: (p − 1, q)(p, q), (p, q)(p + 1, q) ∈ E(C) or (p, q − 1)(p, q), (p, q)(p, q + 1) ∈ E(C). In both cases, let us see that (p, q) is a shortcut vertex. Suppose (p−1, q)(p, q) and (p, q)(p+1, q) are edges in C. It is trivial to see that there exist some q = q such that (p, q ) ∈ C. Then, there is a shortcut σ in C defined by the geodesic in G joining (p, q) and (p, q ). In particular, it is clear that (p, q) is a shortcut vertex since (p, q)(p, q + 1), (p, q)(p, q − 1) / ∈ E(C). A similar argument works when (p, q − 1)(p, q) and (p, q)(p, q + 1) are edges in C.
Case 2. Let us suppose that the three vertices are not aligned. Suppose (p, q − 1)(p, q) and (p, q)(p + 1, q) are edges in C (i. e. (p, q) is an upperleft corner in C). (Any other relative position of the adjacent vertices is equivalent to this one up to rotation of the cycle. Therefore, the argument also works.) In this case, (p − 1, q)(p, q) and (p, q)(p, q + 1) are not edges in C.
Then, consider the graph N and let G be the graph obtained by attaching to each vertex n of N the vertex (0, 0) of G n . Suppose every edge in G has length 1.
Let us see that the resulting graph G is 11-path-chordal. Let C be any cycle in G with L(C) ≥ 11. Clearly, by construction, C is contained in some G n . Let P 1 : G n → {0, 1, ..., n} be such that P 1 (p, q) = p. If P 1 (C) contains more than two points, say p−1, p, p+1, then there exist two vertices (p, q), (p, q ) ∈ C with q = q. Since (p, q − 1)(p, q) ∈ E(G n ) for every 1 ≤ q ≤ n, then the geodesic [(p, q)(p, q )] defines a shortcut. On the other hand, if P 1 (C) contains only two vertices, say p−1, p, since L(C) ≥ 11 there is some q such that (p − 1, q)(p, q) is a shortcut in C.
Then, let us consider the graph N where every edge has length 1 and let G be the graph obtained by attaching to each vertex n in N the vertex (0, 0) of G n .
First, let us see that G is (36, 18)-path-chordal. Let C be any cycle of G with L(C) ≥ 36. Notice that each cycle is contained in some G n . Moreover, it has a vertex v = (p + 1, q + 1) such that (p + 1, q), (p, q + 1) ∈ E(G) and both edges are contained in C (i.e., v is an upper-right vertex of C). Suppose also that p is maximal. By construction, there is an horizontal edge in G n , (p, q )(p + 1, q ), with 0 ≤ q − q ≤ 7. Therefore, [(p, q + 1)(p, q )] ∪ (p, q )(p + 1, q ) defines a path γ in G n with L(γ) < 18. It is immediate to check that this path contains a shortcut σ with L(σ) < 18.
Given any ε > 0 and k ≥ 36 suppose any n > k, 2ε and any vertex v in C n such that v ∈ B C ( n 2 , ε). Since n ≥ 2ε, v = (0, q) for some 0 < q < n. Let us see that (0, q) is not a shortcut vertex.
Suppose σ is a shortcut and (0, q) is a shortcut vertex associated to σ. Since there are no shortcuts in C n from (0, q) to any vertex (p, 0) or (p, n), we may assume that σ ∩ C = {(0, q), (n, q )} for some 0 < q < n. However, by construction of G n , this implies that L(σ) ≥ 4(n − 1) + n > 4n > d C ((0, q), (n, q )), leading to contradiction. Thus, there are no shortcut vertices in B C ( n 2 , ε).  Proof. Suppose that G is ε-densely (k, m)-path-chordal on the triangles. Let us see that δ(G) ≤ max{ k 4 , ε + m}. Consider any geodesic triangle T = {x, y, z}. If L(T ) < k, it follows that every side of the triangle has length at most k 2 . Therefore, the hyperbolic constant is at most k 4 . Then, let L(T ) ≥ k and let us prove that T is (ε + m)-thin. Consider any point p ∈ T and let us assume that p ∈ [xy]. If d(p, x) < ε + m or d(p, y) < ε + m, we are done. Otherwise, there is a shortcut vertex x i such that d(x i , p) < ε and a shortcut σ i , with x i ∈ σ i and L(σ i ) ≤ m. Since [xy] is a geodesic, σ i does not connect two points in [xy] and d(p, [xz] ∪ [yz]) < ε + m.

A characterization of δ-hyperbolic graphs
Suppose that G is δ-hyperbolic and consider any geodesic triangle T = {x, y, z} with L(T ) ≥ 8δ. Let p ∈ T and let us assume, with no loss of generality, that p ∈ [xy]. Since G is δ-hyperbolic, d(p, [xz] ∪ [yz]) < δ. If d(px), d(py) > δ, then there is a path γ with L(γ) < δ joining p to [xz]∪[yz]. In particular, there is a shortcut σ ⊂ γ with L(σ) ≤ L(γ) < δ joining some shortcut vertex p ∈ [xy] with d(p, p ) < δ to [xz] ∪ [yz]. Therefore, if L([x, y]) > 2δ, for every point q ∈ [x, y] there is a shortcut vertex in B T (q, 2δ) ∩ [x, y] associated to a shortcut with length at most δ. Since L(T ) ≥ 8δ, there is at most one side of the triangle with length ≤ 2δ. Then, for every point p in the triangle there is a shortcut vertex in B T (p, 3δ) associated to a shortcut with length at most δ. Thus, it suffices to consider ε = 3δ, k = 8δ and m = δ.