Injective hulls of various graph classes
aa r X i v : . [ c s . D M ] J u l Injective hulls of various graph classes
Heather M. Guarnera ∗ a , Feodor F. Dragan † a , and Arne Leitert ‡ ba Department of Computer Science, Kent State University b Department of Computer Science, Central Washington UniversityJuly 29, 2020
Abstract
A graph is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersectingdisks in G has a common intersection. It is known that for every graph G , there exists a uniquesmallest Helly graph H ( G ) into which G isometrically embeds; H ( G ) is called the injectivehull of G . Motivated by this, we investigate the structural properties of the injective hulls ofvarious graph classes. We say that a class of graphs C is closed under Hellification if G ∈ C implies H ( G ) ∈ C . We identify several graph classes that are closed under Hellification. Weshow that permutation graphs are not closed under Hellification, but chordal graphs, square-chordal graphs, and distance-hereditary graphs are. Graphs that have an efficiently computableinjective hull are of particular interest. A linear-time algorithm to construct the injective hull ofany distance-hereditary graph is provided and we show that the injective hull of several graphsfrom some other well-known classes of graphs are impossible to compute in subexponential time.In particular, there are split graphs, cocomparability graphs, bipartite graphs G such that H ( G )contains Ω( a n ) vertices, where n = | V ( G ) | and a > Keywords: injective hull; Helly graphs; δ -hyperbolic graphs; chordal graphs; square-chordalgraphs; distance-hereditary graphs; permutation graphs. In a graph G = ( V ( G ) , E ( G )), a disk D G ( v, r ) with radius r and centered at a vertex v consists ofall vertices with distance at most r from v , i.e., D G ( v, r ) = { u ∈ V ( G ) : d G ( u, v ) ≤ r } . A graphis called Helly if every system of pairwise intersecting disks has a non-empty common intersection.The injective hull of an arbitrary graph G , denoted H ( G ), is a unique minimal Helly graph whichcontains G as an isometric subgraph [26,31,32]. One measure of how far a graph is from its injectivehull is its Helly-gap , denoted α ( G ), which is the minimum integer k such that every vertex of H ( G )has within distance at most k a vertex of G [21]. Graphs with a small Helly-gap are preciselythe graphs whose disks satisfy the coarse Helly property [9, 21]. As it turns out, many well-knowngraph classes have a small Helly-gap [9, 21] including cube-free median graphs, hereditary modulargraphs, 7-systolic complexes, and the graphs of bounded tree-length, bounded hyperbolicity, orbounded α i -metric.Helly graphs have been well-investigated; they have several characterizations and importantfeatures as established in [4, 5, 17, 18, 20, 33, 35]. They are exactly the so-called absolute retracts of ∗ [email protected] † [email protected] ‡ [email protected] eflexive graphs and possess a certain elimination scheme [4,5,17,18,33] which makes them recogniz-able in O ( n m ) time [17]. The Helly property works as a compactness criterion on graphs [35]. Manynice properties of Helly graphs are based on the eccentricity e G ( v ) of a vertex v , which is defined asthe maximum distance from v to any other vertex of the graph (i.e., e G ( v ) = max u ∈ V ( G ) d G ( v, u )).The minimum and maximum eccentricity in a graph G is the radius and diameter, respectively.Conveniently, the eccentricity function in Helly graphs is unimodal [18], that is, any local minimumcoincides with the global minimum. This fact was recently used in [19, 20, 27] to compute theradius, diameter and a central vertex of a Helly graph in subquadratic time. Helly graphs can bemetrically characterized by the fact that all disks of uniform radius have the Helly property [20].Moreover, there are many graph parameters that are strongly related in Helly graphs, includingso-called interval thinness, hyperbolicity, pseudoconvexity of disks, and size of the largest isometricsubgraph in the form of a square rectilinear grid or a square king grid, among others (cf. [20, 22]);in particular, a constant bound on any one of these parameters implies a constant bound on allothers [20].The rich theory behind Helly graphs entices the use of injective hulls as an underlying structureto solve (approximately) problems on G . Problems such as finding the diameter or computing vertexeccentricities in G are translatable to finding the diameter of H ( G ) and computing eccentricitiesof vertices of the Helly graph H ( G ). Additionally, there is a subquadratic time approximation forradius rad ( G ) of a graph with an additive error depending on α ( G ) [27]. Moreover, the existenceof the injective hull of a graph G is useful to prove properties that appear in G . For example, theexistence of injective hulls has been used to prove the existence of a core which intersects shortestpaths for a majority of pairs of vertices, establishing that traffic congestion is inherent in graphswith global negative curvature [11], a.k.a. hyperbolic graphs. Injective hulls were also used to provethe existence of an eccentricity approximating spanning tree T of G which gives an approximationof all vertex eccentricities with additive error depending essentially on α ( G ) [21].Graph Class C Closed underHellification Hardness to compute H ( G ) for any G ∈ C δ -Hyperbolic Yes Ω( a n )Chordal Yes Ω( a n )Square-Chordal Yes ?Distance-Hereditary Yes O ( n + m )Permutation No ?Cocomparability ? Ω( a n )AT-free ? Ω( a n )Bipartite No Ω( a n )(or any triangle-free)Table 1: A summary of our results on injective hulls of various graph classes, where a > n = | V ( G ) | . ”?” means that this question is still open.The importance of H ( G ) as an underlying structure drives our interest in the injective hulls ofvarious graph classes. Our main contributions are summarized in Table 1 and organized as follows.We identify in Section 3 several universal properties of the injective hull of any graph. Next, wefocus on a graph G that belongs to a particular graph class C . In particular, we are interestedin whether C is closed under Hellification, i.e., whether G ∈ C implies H ( G ) ∈ C . In Section 4,we give a graph theoretic proof that hyperbolic graphs are closed under Hellification. Moreover,we prove that satisfying the Helly property in disks of radii at most δ + 1 is sufficient to satisfythe Helly property in all disks of a δ -hyperbolic graph. In Section 5, we show that permutationgraphs are not closed under Hellification and provide conditions in which AT-free graphs are. In2ection 6, we prove that chordal graphs and square-chordal graphs are closed under Hellification.In Section 7, we add distance-hereditary graphs to the growing list of graph classes closed underHellification and provide a linear-time algorithm to compute H ( G ) of a distance-hereditary graph G . We demonstrate in Section 8 that the injective hull of several graphs from some other well-known classes of graphs are impossible to compute efficiently. Specifically, there is a graph G wherethe number of vertices in H ( G ) is Ω( a n ), where a > n = | V ( G ) | . We construct three such graphs:a split graph, a cocomparability graph, and a bipartite graph. Note also that such well-knowngraph classes as interval graphs, strongly chordal graphs, dually chordal graphs are subclasses ofHelly graphs [7, 17, 25] and therefore for them trivially H ( G ) coincides with G . The definitions ofgraph classes not provided here can be found in [8]. All graphs G = ( V ( G ) , E ( G )) occurring in this paper are undirected, connected, and without loopsor multiple edges. A path P ( v , v k ) is a sequence of vertices v , . . . , v k such that v i v i +1 ∈ E forall i ∈ [0 , k − length is k . The distance d G ( u, v ) between two vertices u and v is the lengthof a shortest path connecting them in G ; the distance d G ( u, S ) between a vertex u and a set ofvertices S ⊆ V ( G ) is the minimum distance from u to any vertex of S . The interval I ( x, y ) betweenvertices x, y is the set of all vertices belonging to a shortest ( x, y )-path, i.e., I ( x, y ) = { v ∈ V ( G ) : d G ( x, y ) = d G ( x, v ) + d G ( v, y ) } . The interval slice S k ( x, y ) is the set of vertices belonging to I ( x, y )and at distance k from x , i.e., S k ( x, y ) = { v ∈ I ( x, y ) : d G ( x, v ) = k } . The neighborhood of v consists of all vertices adjacent to v , denoted by N ( v ). The closed neighborhood of v is definedas N [ v ] = N ( v ) ∪ { v } . The degree deg ( v ) of a vertex v is the number of neighbors it has, i.e., deg ( v ) = | N ( v ) | . A vertex v is pendant if deg ( v ) = 1. Two vertices v and u are twins if theyhave the same neighborhood. True twins are adjacent; false twins are not. A disk D G ( v, r ) withradius r and centered at a vertex v consists of all vertices with distance at most r from v , i.e., D G ( v, r ) = { u ∈ V ( G ) : d G ( u, v ) ≤ r } . A set M ⊂ V ( G ) is said to separate a vertex pair x, y ∈ V ( G ) if the removal of M from G separates x and y into distinct connected components.The eccentricity of a vertex v is defined as e G ( v ) = max u ∈ V ( G ) d G ( v, u ). The radius rad ( G ) anddiameter diam ( G ) are the minimum and maximum eccentricity, respectively. The k th power G k of agraph G is a graph that has the same set of vertices, but in which two distinct vertices are adjacentif and only if their distance in G is at most k . A subgraph G ′ of a graph G is called isometric iffor any two vertices x, y of G ′ , d G ( x, y ) = d G ′ ( x, y ) holds. We denote by h S i the subgraph of G induced by the vertices S ⊂ V . The subindex G is omitted when the graph is known by context.A chord of a path (cycle) v , . . . , v k is an edge between two vertices of the path (cycle) thatis not an edge of the path (cycle). A set M ⊆ V ( G ) is an independent set if for all u, v ∈ V ( G ), uv / ∈ E ( G ). A set M ⊆ V ( G ) is a clique (or complete subgraph ) if all distinct vertices u, v ∈ M have uv ∈ E ( G ). A set M ⊆ V ( G ) is said to be a if for every x, y ∈ M , d ( x, y ) ≤ M is maximal in G if it is maximal by inclusion. A vertex v is said to suspend a set M ⊆ V ( G )if vu ∈ E ( G ) for each u ∈ M \ { v } ; v is also said to be universal to M \ { v } . We denote by C k acycle induced by k vertices, by W k an induced wheel of size k , i.e., a cycle C k with one additionalvertex universal to C k , and by K n a clique of n vertices. A graph B is bipartite if its vertex setcan be partitioned into two independent sets X and Y , i.e., each edge uv ∈ E ( B ) has one end in X and the other in Y .A tree-decomposition ( T , T ) for a graph G is a family T = { B , B , . . . } of subsets of V ( G ),called bags , such that T forms a tree T with the bags in T as nodes which satisfy the followingconditions: (i) each vertex is contained in a bag, (ii) for each edge uv ∈ E ( G ), T has a bag B with u, v ∈ B , and (iii) for each vertex v ∈ V ( G ), the bags containing v induce a subtree of T . A treedecomposition has breadth ρ if, for each bag B , there is a vertex v in G such that B ⊆ D G ( v, ρ ). A3ree decomposition has length λ if the diameter in G of each bag B is at most λ . The tree-breadth tb ( G ) [23] and tree-length tl ( G ) [16] are the minimum breadth and length, respectively, among allpossible tree decompositions of G .A graph G is Helly if, for any system of disks F = { D ( v, r ( v )) : v ∈ S ⊆ V ( G ) } , the followingHelly property holds: if X ∩ Y = ∅ for every X, Y ∈ F , then T v ∈ S D ( v, r ( v )) = ∅ . Pseudo-modular graphs are a far-reaching superclass of Helly graphs. By definition, a graph G is pseudo-modular if every triple x, y, z of its vertices admits either a ‘median’ vertex or a ‘median’ triangle, i.e.,either there is a vertex v such that d ( x, y ) = d ( x, v ) + d ( v, y ) , d ( x, z ) = d ( x, v ) + d ( v, z ) , d ( z, y ) = d ( z, v ) + d ( v, y ) or there is a triangle (three pairwise adjacent vertices) v, u, w such that d ( x, y ) = d ( x, v ) + 1 + d ( u, y ) , d ( x, z ) = d ( x, v ) + 1 + d ( w, z ) , d ( z, y ) = d ( z, w ) + 1 + d ( u, y ). Pseudo-modulargraphs are characterized as follows. Proposition 1. [3] For a connected graph G the following are equivalent:i) G is pseudo-modular.ii) Any three pairwise intersecting disks of G have a nonempty intersection.iii) If ≤ d ( v, w ) ≤ and d ( u, v ) = d ( u, w ) = k ≥ for vertices u, v, w of G , then there exists avertex x such that d ( v, x ) = d ( w, x ) = 1 and d ( u, x ) = k − . The presence of pseudo-modularity in a graph G is of algorithmic interest because it limitsthe number of disk families which must satisfy the Helly property for G to be considered Helly.Specifically, a pseudo modular graph is Helly if and only if it is neighborhood-Helly , i.e., if the familyof its all unit disks (all closed neighborhoods) { D ( v,
1) : v ∈ V ( G ) } satisfies the Helly property. Proposition 2. [4] G is Helly if and only if it is pseudo-modular and neighborhood-Helly. It is clear that G is neighborhood-Helly if and only if all maximal 2-sets of G are suspended.We define the remaining graph classes in their corresponding sections; the definitions of graphclasses not provided here can be found in [8]. By an equivalent definition of an injective hull [26] (also called a tight span), each vertex f ∈ V ( H ( G )) can be represented as a vector with values f ( x ) for each x ∈ V ( G ), such that the followingtwo properties hold: ∀ x, y ∈ V ( G ) f ( x ) + f ( y ) ≥ d ( x, y ) (1) ∀ x ∈ V ( G ) ∃ y ∈ V ( G ) f ( x ) + f ( y ) = d ( x, y ) (2)Additionally, there is an edge between two vertices f, g ∈ V ( H ( G )) if and only if their Chebyshevdistance is 1, i.e., max x ∈ V ( G ) | f ( x ) − g ( x ) | = 1. Thus, d H ( G ) ( f, g ) = max x ∈ V ( G ) | f ( x ) − g ( x ) | . Noticethat if f ∈ V ( H ( G )), then { D ( x, f ( x )) : x ∈ V ( G ) } is a family of pairwise intersecting disks. Fora vertex z ∈ V ( G ), define the distance function d z by setting d z ( x ) = d G ( z, x ) for any x ∈ V ( G ).By the triangle inequality, each d z belongs to V ( H ( G )). An isometric embedding of G into H ( G )is obtained by mapping each vertex z of G to its distance vector d z .We classify every vertex v in V ( H ( G )) as either a real vertex or a Helly vertex. A ver-tex f ∈ V ( H ( G )) is a real vertex provided f = d z for some z ∈ V ( G ), i.e., there is a one-to-onecorrespondence between z ∈ V ( G ) and its representative real vertex f ∈ V ( H ( G )) which uniquelysatisfies f ( z ) = 0 and f ( x ) = d G ( z, x ) for all x ∈ V ( G ). By an abuse of notation, we will inter-changeably use V ( G ) to represent the vertex set in G as well as the vertex subset of H ( G ) whichuniquely corresponds to the vertex set of G . Then, a vertex v ∈ V ( H ( G )) is a real vertex if itbelongs to V ( G ) and a Helly vertex otherwise. Equivalently, a vertex h ∈ V ( H ( G )) is a Hellyvertex provided that h ( x ) ≥ x ∈ V ( G ), that is, a Helly vertex exists only in the injective4ull H ( G ) and not in G . A path P ( x, y ) in H ( G ) connecting vertices x, y ∈ V ( G ) is said to be a real path if each vertex u ∈ P ( x, y ) is real. We often use the terms Hellify (verb) and
Hellification (noun) to describe the process by which edges and Helly vertices are added to G to construct H ( G ).When G is known by context, we often let H := H ( G ).A vertex x is a peripheral vertex if I ( y, x ) I ( y, z ) for some vertex y and all vertices z = x .In H ( G ), all peripheral vertices are real. Consequently, all farthest vertices from any v ∈ V ( H ( G ))are real. It follows that, in H ( G ), any shortest path is a subpath of a shortest path between realvertices. Proposition 3. [21] Peripheral vertices of H ( G ) are real. The following result was proven earlier in [21] only for H := H ( G ). For completeness, we providea proof that it holds for any host H such that G embeds isometrically into H and all peripheralvertices in H are from G . Proposition 4. [21] Let H be a host such that G embeds isometrically into H and all peripheralvertices in H are from G . For any shortest path P ( x, y ) , where x, y ∈ V ( H ) , there is a shortestpath P ( x ∗ , y ∗ ) , where x ∗ , y ∗ ∈ V ( G ) are peripheral vertices of G , such that P ( x ∗ , y ∗ ) ⊇ P ( x, y ) .Proof. If x and y are both real vertices, then the proposition is trivially true. Without loss ofgenerality, suppose vertex y does not belong to V ( G ). Consider a breadth-first search layeringwhere y belongs to layer L i of BFS( H, x ). Let y ′ ∈ L k be a vertex with y ∈ I ( x, y ′ ) that maximizes k = d H ( x, y ′ ). Then, for any vertex z ∈ V ( H ), I ( x, y ′ ) I ( x, z ). Hence, y ′ is a peripheral vertex;by assumption, y ′ ∈ V ( G ). If x / ∈ V ( G ), then applying the previous step using BFS( H, y ′ ) yieldsvertex x ′ ∈ V ( G ).Let the distance d ( z, P ) from a vertex z to an ( x, y )-path P be the minimum distance from z toany vertex u ∈ P . We next show that for any vertex z ∈ V ( H ( G )) and any ( x, y )-path P in H ( G ),there is a real ( x ∗ , y ∗ )-path P ∗ in G which behaves similarly to P with respect to some distanceproperties. In particular, we show that if x, y, z ∈ V ( G ) then for every ( x, y )-path P in H ( G ), thereis a real ( x, y )-path P ∗ in G such that d ( z, P ∗ ) ≥ d ( z, P ).We have the following lemma. Lemma 1.
Let H be the injective hull of G . For any vertex z ∈ V ( H ) and edge xy ∈ E ( H ) ,there is a real ( x ∗ , y ∗ ) -path P ∗ in G such that d H ( z, P ∗ ) ≥ d H ( z, { x, y } ) , I ( z, x ) ⊆ I ( z, x ∗ ) , and I ( z, y ) ⊆ I ( z, y ∗ ) .Proof. Let L , L , . . . , L e ( z ) be layers of H produced by a breadth-first search rooted at vertex z .Without loss of generality, let d H ( z, { x, y } ) = d H ( z, x ) = k . Hence, x ∈ L k and y ∈ L p where p = k or p = k + 1. By Proposition 4, there is a vertex x ∗ ∈ V ( G ) such that x ∈ I ( z, x ∗ ) and there is avertex y ∗ ∈ V ( G ) such that y ∈ I ( z, y ∗ ). Then, x ∗ ∈ L j for some j ≥ k and y ∗ ∈ L ℓ for some ℓ ≥ p .Since G is isometric in H , there is a shortest ( x ∗ , y ∗ )-path P ∗ in G of length d H ( x ∗ , y ∗ ) consisting ofall real vertices, as illustrated in Figure 1(a). By the triangle inequality, d G ( x ∗ , y ∗ ) = d H ( x ∗ , y ∗ ) ≤ d H ( x ∗ , x ) + 1 + d H ( y, y ∗ ) ≤ ( j − k ) + 1 + ( ℓ − p ). By contradiction, assume there is a vertex w ∗ ∈ P ∗ such that d H ( z, w ∗ ) < k . Then, d H ( x ∗ , y ∗ ) = d H ( x ∗ , w ∗ ) + d H ( w ∗ , y ∗ ) ≥ ( j − k + 1) + ( ℓ − k + 1),a contradiction. Theorem 1.
Let H be the injective hull of G . For any x, y, z ∈ V ( G ) , the disk D G ( z, k ) separatesvertices x, y in G if and only if disk D H ( z, k ) separates vertices x, y in H .Proof. ( ← ) It suffices to remark that if D G ( z, k ) does not separate x, y in G due to a path P ∗ connecting them, then the same path establishes that D H ( z, k ) does not separate x, y in H ( G ).5 L k L j L l zx yx ∗ y ∗ P ∗ (a) zx yx ∗ y ∗ d H ( z, P ) > kP P ∗ (b) Figure 1: Illustration to the proofs of (a) Lemma 1 and (b) Theorem 1, where real paths are shownin blue.( → ) Suppose the disk D H ( z, k ) does not separate vertices x, y in H and assume, without loss ofgenerality, that D H ( z, k ) ∩ { x, y } = ∅ . Then, there is an ( x, y )-path P in H such that d H ( z, P ) > k .Let P = v , v , v , . . . , v j , where v := x and v j := y . By Lemma 1, for each edge v i v i +1 on P ,there is a real ( v i ∗ , v i +1 ∗ )-path P i ∗ in G such that d H ( z, P i ∗ ) ≥ d H ( z, { v i , v i +1 } ) > k , as shownin Figure 1(b). Let P ∗ be the real path obtained by joining, for i ∈ [0 , j − P ∗ i by their end vertices. Then, d H ( z, P ∗ ) ≥ d H ( z, P ) > k . As a result, the disk D G ( z, k ) does notseparate vertices x, y in G . Corollary 1.
Let H be the injective hull of G . For any x, y, z ∈ V ( G ) and every ( x, y ) -path P in H , there is a real ( x, y ) -path P ∗ in G such that d ( z, P ∗ ) ≥ d ( z, P ) . δ -Hyperbolic graphs A metric space (
X, d ) is δ -hyperbolic if it satisfies Gromov’s 4-point condition: for any four points u, v, w, x from X the two larger of the three distance sums d ( u, v ) + d ( w, x ), d ( u, x ) + d ( v, w ),and d ( u, w ) + d ( v, x ) differ by at most 2 δ ≥
0. A connected graph equipped with the standardgraph metric d G is δ -hyperbolic if the metric space ( V, d G ) is δ -hyperbolic. The smallest value δ for which G is δ -hyperbolic is called the hyperbolicity of G and is denoted δ ( G ). Note that δ ( G ) isan integer or a half-integer. For a quadruple of vertices u, v, w, x ∈ V ( G ), it will be convenient todenote by hb ( u, v, w, x ) half the difference of the largest two distance sums among d ( u, v ) + d ( w, x ), d ( u, x ) + d ( v, w ), and d ( u, w ) + d ( v, x ).It is known [31,32] that the hyperbolicity of any metric space is preserved in its injective hull. Forcompleteness, we provide a graph-theoretic proof of this result and show that in fact hyperbolicityis preserved in any host H as long as distances in G are preserved in H and that peripheral verticesof H are real. Proposition 5. If H is a host graph such that G embeds isometrically into H and all peripheralvertices in H are from G , then δ ( G ) = δ ( H ) .Proof. As G embeds isometrically into H , δ ( G ) ≤ δ ( H ). By contradiction, assume δ ( H ) > δ ( G ).Let x, y, z, t ∈ V ( H ) with hb ( x, y, z, t ) > δ ( G ) such that | V ( G ) ∩ { x, y, z, t }| is maximized. Withoutloss of generality, let d H ( x, t ) + d H ( z, y ) ≥ d H ( x, z ) + d H ( t, y ) ≥ d H ( x, y ) + d H ( z, t ). If { x, y, z, t } ⊆ V ( G ), then hb ( x, y, z, t ) ≤ δ ( G ), a contradiction. Thus, without loss of generality, suppose x / ∈ V ( G ). By Proposition 4, there is a peripheral vertex x ∗ ∈ V ( G ) such that I ( t, x ) ⊂ I ( t, x ∗ ) for6ertex t ∈ V ( H ). Let d H ( t, x ∗ ) = d H ( t, x ) + γ . Clearly, d H ( x ∗ , t ) + d H ( z, y ) ≥ max { d H ( x ∗ , y ) + d H ( z, t ) , d H ( x ∗ , z ) + d H ( t, y ) } .Suppose that d H ( x ∗ , y )+ d H ( z, t ) ≥ d H ( x ∗ , z )+ d H ( t, y ). By the triangle inequality and definitionof hyperbolicity, we have2 hb ( x ∗ , y, z, t ) = d H ( x ∗ , t ) + d H ( z, y ) − d H ( x ∗ , y ) − d H ( z, t ) ≥ d H ( x, t ) + d H ( z, y ) + γ − d H ( x, y ) − d H ( z, t ) − γ = d H ( x, t ) + d H ( z, y ) − d H ( x, y ) − d H ( z, t ) ≥ d H ( x, t ) + d H ( z, y ) − d H ( x, z ) − d H ( t, y )= 2 hb ( x, y, z, t ) . Thus, hb ( x ∗ , y, z, t ) ≥ hb ( x, y, z, t ), a contradiction with the maximality of the number of realvertices in the quadruple.Suppose now that d H ( x ∗ , z ) + d H ( t, y ) ≥ d H ( x ∗ , y ) + d H ( z, t ). By the triangle inequality anddefinition of hyperbolicity, we have2 hb ( x ∗ , y, z, t ) = d H ( x ∗ , t ) + d H ( z, y ) − d H ( x ∗ , z ) − d H ( t, y ) ≥ d H ( x, t ) + d H ( z, y ) + γ − d H ( x, z ) − d H ( t, y ) − γ = d H ( x, t ) + d H ( z, y ) − d H ( x, z ) − d H ( t, y )= 2 hb ( x, y, z, t ) . Thus, hb ( x ∗ , y, z, t ) ≥ hb ( x, y, z, t ), again a contradiction with the maximality of the number of realvertices in the quadruple. Theorem 2.
For any graph G , δ ( G ) = δ ( H ( G )) . That is, the δ -hyperbolic graphs are closed underHellification. We next show that a δ -hyperbolic graph G is Helly if its disks up to radii δ + 1 satisfy the Hellyproperty. In this sense, a localized Helly property implies a global Helly property, akin to what isknown for pseudo-modular graphs wherein all disks of radii at most 1 satisfy the Helly propertyimplies all disks (of all radii) satisfy the Helly property. Lemma 2. If G is δ -hyperbolic and all disks with up to δ + 1 radii satisfy the Helly property, then G is a Helly graph.Proof. Assume all disks with radii at most δ + 1 satisfy the Helly property. Clearly G isneighborhood-Helly. By Proposition 2, it remains only to prove that G is pseudo-modular. Weapply Proposition 1(iii). Consider three vertices u, v, w such that d ( u, v ) = d ( u, w ) = k ≥
2, andeither v and w are adjacent or have a common neighbor z . We claim that d ( u, v ) = d ( u, w ) = k implies there is a vertex t adjacent to v and w and at distance k − u . We use an inductionon d ( u, v ). By assumption, it is true for k ≤ δ + 2 as the pairwise-intersecting disks D ( u, k − D ( v, D ( w,
1) have a common vertex t by the Helly property.Consider the case when d ( u, v ) = d ( u, w ) = k > δ + 2. Let x ∈ I ( v, u ) and y ∈ I ( w, u ) bevertices such that d ( x, u ) = d ( y, u ) = δ + 2. We claim the disks D ( x, δ + 1), D ( y, δ + 1), and D ( u,
1) pairwise intersect; then, vertex u ∗ exists by the Helly property and applying the inductivehypothesis to vertex u ∗ equidistant to v, w yields the desired vertex t . Clearly, D ( u,
1) intersectsboth D ( x, δ + 1) and D ( y, δ + 1). It remains to show that d ( x, y ) ≤ δ + 2.Consider vertices u, x, y, w and three distance sums: A := d ( u, w )+ d ( x, y ), B := d ( u, y )+ d ( x, w )and C := d ( u, x ) + d ( y, w ). We have A = k + d ( x, y ) and C = k . Moreover, k ≤ B ≤ k + 2 as k = d ( u, w ) ≤ d ( u, x ) + d ( x, v ) + d ( v, w ) ≤ d ( u, v ) + 2 = k + 2. Hence, C is a smallest sum. If B ≥ A then k + 2 ≥ B ≥ A = k + d ( x, y ) implies d ( x, y ) ≤ ≤ δ + 2. If A ≥ B then, by 4-pointcondition, 2 δ ≥ A − B ≥ k + d ( x, y ) − k −
2, i.e., d ( x, y ) ≤ δ + 2.7 Permutation graphs and relatives
Permutation graphs can be defined as follows. Consider two parallel lines (upper and lower) in theplane. Assume that each line contains n points, labeled 1 to n , and each two points with the samelabel define a segment with that label. The intersection graph of such a set of segments betweentwo parallel lines is called a permutation graph [8]. An asteroidal triple is an independent set ofthree vertices such that each pair is joined by a path that avoids the closed neighborhood of thethird. A far reaching superclass of permutation graphs are the AT-free graphs, i.e., the graphs thatdo not contain any asteroidal triples [12].We show that permutation graphs are not closed under Hellification. Moreover, if the Helly-gapof some AT-free graph is 2, then AT-free graphs are also not closed under Hellification. ab cd ef (a) b da cf e (b) b da cf eh h (c) Figure 2: A permutation model (a) corresponding to permutation graph G (b) and its injectivehull H ( G ) (c), where H ( G ) is not a permutation graph. Lemma 3.
Permutation graphs are not closed under Hellification.Proof.
The graph G illustrated in Figure 2 is an example of a permutation graph G for which H ( G )is not a permutation graph (although, H ( G ) is AT-free). Note that only two Helly vertices h and h are added to produce H ( G ), where h is adjacent to real vertices b, d, a, c, f and h e / ∈ E ( H ( G )).The resulting graph H ( G ) is not a permutation graph since such a vertex/segment h cannot beadded to the essentially unique permutation model of G depicted in Figure 2; any segment h intersecting the segments b, d, a, c, f needs to intersect also the segment e .For an AT-free graph G , the Helly-gap α ( G ) is impacted by whether H ( G ) is AT-free. Recallthat the Helly gap α ( G ) is the minimum integer α such that the distance from any Helly vertex h ∈ V ( H ) to a closest real vertex x ∈ V ( G ) is at most α . It is known [21] that any AT-free graph G has α ( G ) ≤ Lemma 4.
For any graph G , α ( G ) ≤ if H ( G ) is AT-free.Proof. By contradiction, suppose α ( G ) ≥ G and H := H ( G ) is AT-free. Then,there is a vertex h ∈ V ( H ) such that d H ( h, v ) ≥ α ( G ) for all v ∈ V ( G ). Let x ∈ V ( G ) beclosest to h ; then, d H ( h, x ) = α ( G ) ≥
2. By Proposition 4, there is a real vertex y ∈ V ( G ) suchthat h ∈ I ( x, y ). Moreover, d H ( h, y ) ≥ d H ( h, x ) ≥
2. Let P be a shortest ( x, y )-path of H with h ∈ P . As G is isometric in H , there is a (real) shortest ( x, y )-path P ∗ in G . By d H ( x, y ) distancerequirements, all shortest ( h, y )-paths avoid N [ x ], and all shortest ( h, x )-paths avoid N [ y ]. As P ∗ ⊆ V ( G ) and α ( G ) ≥
2, then P ∗ also avoids N [ h ]. Therefore, { x, y, h } forms an asteroidal triplein H , a contradiction. Corollary 2.
If there is an AT-free graph G with α ( G ) = 2 , then AT-free graphs are not closedunder Hellification. Currently, we do not know whether there is an AT-free graph G with α ( G ) = 2.8 Chordal Graphs and Square-Chordal Graphs
A graph is chordal if it contains no induced cycle C k of length k ≥
4. A graph G is square-chordal if G is chordal. In this section, we will show that for a chordal (square-chordal) graph G , itsinjective hull H ( G ) is also chordal (square-chordal). That is, chordal graphs and square-chordalgraphs are closed under Hellification.The following fact is a folklore. Proposition 6.
Let G be a chordal graph, and let C be a cycle of G . For any vertex x ∈ C , if x is not adjacent to any third vertex of C , then the neighbors in C of x are adjacent. We will need a few auxiliary lemmas. The following characterizations of chordal graphs withinthe class of the α -metric graphs will be useful. A graph is said to be an α -metric graph if itsatisfies the following: for any x, y, z, v ∈ V ( G ) such that zy ∈ E ( G ), z ∈ I ( x, y ) and y ∈ I ( z, v ), d G ( x, v ) ≥ d G ( x, y ) + d G ( y, v ) − Lemma 5. [38] G is a chordal graph if and only if it is an α -metric graph not containing anyinduced subgraphs isomorphic to cycle C and wheel W k , k ≥ . A graph is bridged [28] if it contains no isometric cycle C k of length k ≥
4. Bridged graphs area natural generalization of chordal graphs. Directly combining two results from [24, 38], we obtainthe following lemma.
Lemma 6. [24, 38] G is an α -metric graph not containing an induced C if and only if G is abridged graph not containing W ++6 as an isometric subgraph (see Figure 3). Figure 3: Forbidden isometric subgraph W ++6 Next lemma establishes conditions in which a Helly graph is chordal.
Lemma 7. If G is a Helly graph with no induced wheels W k , k ≥ , then G is chordal.Proof. We first claim that G has no induced C nor C . By contradiction, assume C or C isinduced in G . Consider the system of pairwise intersecting unit disks centered at each vertex ofthe cycle. By the Helly property, there is a vertex universal to the cycle. Thus, G contains W or W , a contradiction establishing the claim that G has no induced C nor C .We next claim that G is a bridged graph. Suppose G has an isometric cycle C ℓ for someinteger ℓ ≥ ℓ = 2, G has induced C ). Let x, y ∈ C ℓ be opposite vertices such that d G ( x, y ) = ℓ . Let z, t ∈ C ℓ be the distinct neighbors of y , as illustrated in Figure 4(a). As thedisks D ( x, ℓ − D ( z, D ( t,
1) pairwise intersect, then by the Helly property, there is avertex v ∈ I ( x, t ) ∩ I ( x, z ) ∩ I ( t, z ). Since C ℓ is isometric, necessarily vy / ∈ E ( G ) and zt / ∈ E ( G ).A contradiction arises with the C induced by v, z, y, t .Suppose now that G has an isometric cycle C ℓ +1 for some integer ℓ ≥ ℓ = 2, G hasinduced C ). Let x, y , y ∈ C ℓ +1 be vertices such that y y ∈ E ( G ) and d G ( x, y ) = d G ( x, y ) = ℓ .As the disks D ( x, ℓ − D ( y , D ( y ,
1) pairwise intersect, by the Helly property, there isa vertex v adjacent to y and y such that d G ( x, v ) = ℓ −
1. Let z, t ∈ C ℓ +1 be vertices such that z ∈ N ( y ) ∩ I ( y , x ) and t ∈ N ( y ) ∩ I ( y , x ), as illustrated in Figure 4(b). Since ℓ ≥ z, t on isometric cycle C ℓ +1 , necessarily d G ( z, t ) = 3. Therefore, vz / ∈ E ( G )or vt / ∈ E ( G ); without loss of generality, let vz / ∈ E ( G ). As the disks D ( x, ℓ − D ( v, D ( z,
1) pairwise intersect, by the Helly property, there is a vertex u ∈ I ( x, v ) ∩ I ( x, z ) ∩ I ( v, z ).Necessarily uy / ∈ E ( G ), otherwise d G ( x, y ) < ℓ . A contradiction arises with the C induced by u, z, y , v .Hence, G is a bridged graph. Since G has no induced W k for k ≥ G does not contain W ++6 as an isometric subgraph (observe that W is an isometric subgraph of W ++6 ). By Lemma 6, G isan α -metric graph not containing an induced C . Since G also has no induced W k for k ≥
4, byLemma 5, G is chordal. x yztv ℓ − ℓ − ℓ − C ℓ x y y ztvu ℓ − ℓ − ℓ − ℓ − (b) Case of isometric C ℓ +1 Figure 4: Illustration to the proof of Lemma 7.We will also use the fact that chordal graphs and square-chordal graphs can be characterizedby the chordality of their so-called visibility graph and intersection graph, respectively. Let M = { S , . . . , S ℓ } be a family of subsets of V ( G ), i.e., each S i ⊆ V ( G ). An intersection graph L ( M ) anda visibility graph Γ( M ) are both a generalization of graph powers and are defined by Brandst¨adt etal. [6] as follows. The sets from M are the vertices of L ( M ) and Γ( M ). Two vertices of L ( M ) arejoined by an edge if and only if their corresponding sets intersect. Two vertices of Γ( M ) are joinedby an edge if and only if their corresponding sets are visible to each other; two sets S i and S j arevisible to each other if S i ∩ S j = ∅ or there is an edge of G with one end in S i and the other end in S j . Denote by D ( G ) = { D ( v, r ) : v ∈ V ( G ) , r a non-negative integer } the family of all disks of G . Lemma 8. [6] For a graph G , Γ( D ( G )) is chordal if and only if G is chordal. Lemma 9. [6] For a graph G , L ( D ( G )) is chordal if and only if G is chordal. We are now ready to prove the main results of this section.
Theorem 3.
Let G be a chordal graph. Then H ( G ) is also chordal.Proof. By contradiction, assume G is chordal and H := H ( G ) is not. By Lemma 7, there is aninduced wheel W k in H for some k ≥
4. Let S = { v , . . . , v k } be the set of vertices of W k thatinduce a cycle C k suspended by universal vertex c .We first claim that there is a real vertex u such that d H ( u , v i ) = d H ( u , v )+ d H ( v , v i ) for each v i ∈ S . Consider the layering L , . . . , L λ produced by a multi-source breadth-first search rootedat the vertex set { v , v , . . . , v k } ; this can be simulated with a BFS rooted at an artificial vertex s adjacent to only { v , v , . . . , v k } . Then, L = { s } , L = { v , . . . , v k } , { v , v , c } ⊆ L , and v ∈ L .Let vertex u be a vertex in L ρ such that ρ is maximal and d H ( v , u ) = ρ − v to u intersects each layer only once); then, d H ( u , v i ) = d H ( u , v ) + d H ( v , v i )holds for each v i ∈ S . By maximality of ρ , there is no vertex z ∈ V ( H ) with I ( v , u ) ⊂ I ( v , z ).Therefore, u is a peripheral vertex and, by Proposition 3, is real (see Figure 5).For each remaining vertex v i ∈ S , we define a corresponding real vertex u i in the following way.By Proposition 4, there are two real vertices u , u such that a shortest path between them contains10 ( v , v ) = v cv as a subpath. Thus, d H ( u , u ) = d H ( u , v ) + 2 + d H ( v , u ). Now let j ∈ [4 , k ]be an integer. By choice of u , vertices c and v belong to I ( u , v j ). Denote by P ( u , v j ) a shortestpath containing c, v . By Proposition 4, there is a (not necessarily distinct) real vertex u j suchthat shortest path P ( u , u j ) contains P ( u , v j ). Thus, d H ( u , u j ) = d H ( u , v ) + 2 + d H ( v j , u j ).With all distances established, we consider in G the family of disks { D ( u i , r ( u i )) } , where r ( u i ) = d H ( u i , v i ), for each v i ∈ S . The disks centered at each vertex u i ∈ V ( G ) are visible to each other iftheir corresponding vertices v i ∈ V ( H ) are adjacent, i.e., d G ( u i , u j ) ≤ d H ( u i , v i ) + 1 + d H ( v j , u j ) = r ( u i ) + r ( v i ) + 1 if v i v j ∈ E ( H ). As d H ( u , u ) = r ( u ) + r ( u ) + 2, the disk D ( u , r ( u )) anddisk D ( u , r ( u )) are not visible to each other. As d H ( u , u j ) = r ( u ) + r ( u j ) + 2, for each integer j ∈ [4 , k ], the disk D ( u , r ( u )) is not visible to the disk D ( u j , r ( u j )). Consider the visibilitygraph Γ( D ( G )). The vertices D ( u i , r ( u i )) ∈ V (Γ( D ( G ))), i ∈ { , . . . , k } , form a cycle in Γ( D ( G )).As vertex D ( u , r ( u )) is not adjacent to any vertex D ( u j , r ( u j )), where j ∈ { , . . . , k } , and itsneighbors D ( u , r ( u )) and D ( u , r ( u )) on the cycle are not adjacent, by Proposition 6, Γ( D ( G ))is not chordal. By Lemma 8, G is also not chordal, a contradiction. . . . . . . L L L L L ρ L λ v v v v v v k c u s Figure 5: Illustration to the proof of Theorem 3.A similar proof shows that the injective hull of a square-chordal graph G is also square-chordal.We will need the following lemma. Lemma 10. [17] Any power of a Helly graph is also a Helly graph.
Theorem 4. If G is square-chordal, then H ( G ) is square-chordal.Proof. Let H := H ( G ). By Lemma 10, H is Helly. Assume, by contradiction, that G is chordalbut H is not. By Lemma 7, there is an induced wheel W k in H for some k ≥
4. Let S = { v , . . . , v k } be the set of vertices of W k that induce a cycle C k suspended by universal vertex c . As cv i ∈ E ( H ) for each v i ∈ S , then d H ( c, v i ) ≤
2. We denote by v z a particular vertex of S definedas follows. If there is a vertex v i ∈ S such that cv i ∈ E ( H ), then set v z := v i . In this case, observethat all vertices v j ∈ S \ D H ( v i ,
1) satisfy d H ( v j , c ) = 2, else S would not induce an induced cyclein H . As k ≥
4, there is at least one such vertex v j . On the other hand, if d H ( c, v i ) = 2 for each v i ∈ S , then let v z be any vertex of S . Without loss of generality, in what follows, we can assumethat v z is v .In the next few steps, we define for each vertex v i ∈ S a real vertex u i satisfying particulardistance requirements. By Proposition 4, there are real vertices u , u ∈ V ( G ) such that a shortest( u , u )-path in H contains a shortest ( v , v )-path in H . As v and v are non-adjacent in H , d H ( v , v ) ≥ d H ( u , u ) ≥ d H ( u , v )+ d H ( v , u )+3. Consider now a multi-sourcebreadth-first search in H rooted at M = S \ { v , v , v } ; this can be simulated with a BFS rooted atan artificial vertex s adjacent to only the vertices of M . Then, L = { s } , L = M , v , v ∈ L ∪ L , c ∈ L , and finally, v ∈ L µ for µ = 4 or µ = 5. Let vertex u be a vertex in L ρ such that ρ ismaximal and d H ( v , u ) = ρ − µ (i.e., each shortest path from v to u intersects each layer only11nce); By maximality of ρ , there is no vertex f ∈ V ( H ) with I ( v , u ) ⊂ I ( v , f ). Therefore, u isa peripheral vertex and, by Proposition 3, is real.For each remaining vertex v i ∈ M , we define a corresponding real vertex u i in the following way.Note that, by choice of v i , v and v i are non-neighbors in H ; thus, d H ( v i , v ) ≥
3. On one hand,if v ∈ I ( v i , u ) then, by Proposition 4, there is a real u i vertex such that a shortest ( u i , u )-pathin H contains a shortest ( v i , v )-path in H . Hence, d H ( u i , u ) ≥ d H ( u i , v i ) + d H ( v , u ) + 3. Onthe other hand, if v / ∈ I ( v i , u ), then necessarily v ∈ L and d H ( v i , v ) = 4. Then, there exists avertex z ∈ I ( v i , u ) ∩ L with d H ( v i , z ) = 3 and d H ( z, u ) = d H ( v , u ). By Proposition 4, there isa real vertex u i such that a shortest ( u i , u )-path in H contains a shortest ( v i , z )-path in H . Hence, d H ( u i , u ) = d H ( u i , v i ) + d H ( v i , z ) + d H ( z, u ) = d H ( u i , v i ) + 3 + d H ( v i , u ).With all distances established, we consider in G the family of disks { D ( u i , r ( u i )) } , where r ( u i ) = d H ( u i , v i ) + 1, for each v i ∈ S . The disks centered at each vertex u i ∈ V ( G ) intersect if theircorresponding vertices v i ∈ V ( H ) are adjacent in H , i.e., d G ( u i , u j ) ≤ d H ( u i , v i )+2+ d H ( v j , u j ) = r ( u i ) + r ( u j ) if v i v j ∈ E ( H ). As d H ( u , u ) ≥ r ( u ) + r ( u ) + 1, the disk D ( u , r ( u )) and disk D ( u , r ( u )) do not intersect. As d H ( u , u j ) ≥ r ( u ) + r ( u j ) + 1, for each j ∈ { , . . . , k } , thedisks D ( u , r ( u )) and D ( u j , r ( u j )) do not intersect. Consider the intersection graph L ( D ( G )).The vertices D ( u i , r ( u i )) ∈ V ( L ( D ( G ))), i ∈ { , . . . , k } , form a cycle in L ( D ( G )). As vertex D ( u , r ( u )) is not adjacent to any vertex D ( u j , r ( u j )), where j ∈ { , . . . , k } , and its neighbors D ( u , r ( u )) and D ( u , r ( u )) on the cycle are not adjacent, by Proposition 6, L ( D ( G )) is notchordal. By Lemma 9, G is also not chordal, a contradiction.A graph G is dually chordal if it has a so-called maximum neighborhood ordering (see [7, 25] fordefinitions and various characterizations of this class of graphs). A maximum neighborhood orderingcan be constructed in total linear time [7,25]. For us here, the following characterization is relevant:a graph G is dually chordal if and only if G is neighborhood-Helly and G is chordal [7, 25]. So, wecan state the following corollary. Corollary 3. If G is a square-chordal graph, then H ( G ) is dually chordal. A graph is distance-hereditary if and only if each of its connected induced subgraphs is isomet-ric [30], that is, the length of any induced path between two vertices equals their distance in G .In this section, we show that distance-hereditary graphs are closed under Hellification. We give acharacterization of the distance-hereditary Helly graphs and show conditions under which addinga vertex to a Helly graph keeps it Helly in Section 7.1. In Section 7.2 we describe a data structurewhich we then use in Section 7.3 to construct the injective hull of a distance-hereditary graph inlinear time.We use the following characterizations of distance-hereditary graphs. Proposition 7. [2, 15] For a graph G , the following conditions are equivalent:(i) G is distance-hereditary;(ii) The house, domino, gem, and the cycles C k of length k ≥ are not induced subgraphs of G (see Figure 6);(iii) G is obtained from K by a sequence of one-vertex extensions: attaching a pendant vertex ora twin vertex. Let w, x, y, z be four vertices that induce a C . We denote by S ( w, x, y, z ) an extended squarethat includes the vertices w, x, y, z which induce a C and any vertex adjacent to at least three12 ouse Domino
Gem
Figure 6: Forbidden induced subgraphs in a distance-hereditary graph.of them, i.e., S ( w, x, y, z ) = { v ∈ D ( { w, x, y, z } ,
1) : | N [ v ] ∩ { w, x, y, z }| ≥ } . We show that adistance-hereditary graph is Helly if and only if all extended squares are suspended. The result isanalogous to a characterization of chordal Helly graphs [17]: a chordal graph is Helly if and only ifall extended triangles are suspended, where an extended triangle S ( x, y, z ) is defined as the set ofvertices that see at least two vertices of the triangle ∆( x, y, z ). Lemma 11.
A distance-hereditary graph G is Helly if and only if, for every C induced by w, x, y, z ∈ V ( G ) , the extended square S ( w, x, y, z ) is suspended.Proof. It is known that distance-hereditary graphs are pseudo-modular [3]. Hence, by Proposition 2, G is Helly if and only if it is neighborhood-Helly, i.e., all 2-sets are suspended. If G is neighborhood-Helly, then all extended squares are suspended since all vertices of an extended square are pairwiseat distance at most 2. We assert that if every extended square is suspended, then all 2-sets aresuspended.We use an induction on the cardinality of a 2-set. Assume any 2-set M ⊆ V ( G ) with | M | ≤ k is suspended. Clearly, it is true for k ≤
2. By contradiction, assume every extended square issuspended but there is an unsuspended 2-set M with | M | = k + 1 ≥
3. Let v , v , v ∈ M . By theinductive hypothesis, there is a vertex c universal to M \ { v } , a vertex c universal to M \ { v } ,and a vertex c universal to M \ { v } . Since M is not suspended, necessarily each c i ∈ { c , c , c } has c i v i / ∈ E ( G ), c i = v i , and c i is distinct from the other two vertices of { c , c , c } . We considerthree cases based on how many of { c , c , c } are distinct from the three vertices { v , v , v } . Wewill obtain a forbidden induced subgraph which contradicts Proposition 7(ii) or will show that M is a subset of some extended square which is suspended, giving a contradiction with M beingunsuspended. Case 1. c / ∈ { v , v , v } and c , c ∈ { v , v , v } . Without loss of generality, let c = v . If c = v , then v c ∈ E ( G ), i.e., c v ∈ E ( G ), a contradic-tion. Hence, c = v and therefore, c , c , c , v induce C . Any x ∈ M belongs to S ( c , c , c , v )since x is either one of c , c , v or adjacent to all of c , c , c . Thus, M ⊆ S ( c , c , c , v ). Case 2. c , c / ∈ { v , v , v } and c ∈ { v , v , v } . Without loss of generality, let c = v . Then, c is adjacent to v , but c v / ∈ E ( G ). Since c v ∈ E ( G ) and c v / ∈ E ( G ), by equality c c ∈ E ( G ) and c c / ∈ E ( G ). By assumption, c is adjacent to v and v , c is adjacent to v , and c v / ∈ E ( G ). It only remains whether c c ∈ E ( G ) and/or v v ∈ E ( G ). If at most one of those edges occurs, we obtain C or a houseinduced by { c , c , v , c , v } . Therefore, both edges c c and v v must be present. Now, any x ∈ M \ { c , v , v } is adjacent to both c , c . Furthermore, if xv / ∈ E ( G ) and xv / ∈ E ( G ), then x, c , c , v , v induce a house. Thus, x is also adjacent to at least one of v , v . Since any x ∈ M is either one of v , v or is adjacent to at least three of { c , c , v , v } , we get M ⊆ S ( c , c , v , v ). Case 3. c , c , c / ∈ { v , v , v } . By assumption, c is adjacent to v and v , c is adjacent to v and v , c is adjacent to v and v ,and c v , c v , c v / ∈ E ( G ). If each i, j ∈ { , , } with i = j satisfies c i c j / ∈ E ( G ) and v i v j / ∈ E ( G ),then v , c , v , c , v , c induce C . Thus, there is some chord c i c j ∈ E ( G ) or v i v j ∈ E ( G ). Weconsider two subcases without loss of generality. Case 3(a). There is a chord c c ∈ E ( G ) .
13f there are no other edges between vertices { c , c , v , c , v } , then those vertices induce a C . Thus,there is at least one of the following chords: c c , v v , or c c . If v v / ∈ E ( G ), we get in G a houseor gem induced by v , c , c , v , c . Hence, v v ∈ E ( G ). Consider now C induced by c , c , v , v .Any vertex x ∈ M \ { c , c , v , v } is adjacent to both c , c . Furthermore, if xv / ∈ E ( G ) and xv / ∈ E ( G ), then x, c , c , v , v induce a house. Thus, x is also adjacent to at least one of v , v .Since any x ∈ M is either one of c , c , v , v or is adjacent to at least three of { c , c , v , v } , weget M ⊆ S ( c , c , v , v ). Case 3(b). There is a chord v v ∈ E ( G ) and c i c j / ∈ E ( G ) for distinct i, j ∈ { , , } . If there are no other edges between vertices { v , c , v , c , v } , then those vertices induce C . Thus,there is at least one of the following chords: v v or v v . But then, vertices v , v , c , v , c inducea house or a gem. Obtained contradictions prove the lemma.We found it advantageous to use a characteristic pruning sequence of G (see Proposition 7(iii)).A pruning sequence σ G : V ( G ) → { , . . . , n } of G is a total ordering of its vertex set V ( G ) = { v , . . . , v n } such that each vertex v i satisfies one of the following conditions in the induced subgraph G i := h v , . . . , v i i :(i) v i is a pendant vertex to some vertex v j with σ G ( v j ) < σ G ( v i ),(ii) v i is a true twin of some vertex v j with σ G ( v j ) < σ G ( v i ), or(iii) v i is a false twin of some vertex v j with σ G ( v j ) < σ G ( v i ).Next lemmas give conditions under which adding a vertex to a Helly graph keeps it Helly.Consider a graph H obtained by adding to a Helly graph G a vertex u as a pendant or twin tosome vertex in G . We show that any family F = (cid:8) D H ( w, r ( w )) : w ∈ M ⊆ V ( H ) (cid:9) of pairwiseintersecting disks in H has a common intersection. Note that this is trivially true if any vertex w ∈ M has r ( w ) = 0 (since w is common to all disks of F ) or if u / ∈ M (since G is isometric in H and the family of pairwise intersecting disks (cid:8) D G ( w, r ( w )) : w ∈ M ⊆ V ( G ) (cid:9) have a commonintersection in G ). Lemma 12.
Let G + { u } be a graph obtained by adding a vertex u pendant to v ∈ V ( G ) . If G isHelly, then G + { u } is Helly.Proof. Let H := G + { u } , F = (cid:8) D H ( w, r ( w )) : w ∈ M ⊆ V ( H ) (cid:9) be a family of pairwise intersectingdisks in H , and u ∈ M . If r ( u ) ≥
2, one may substitute in F the disk D H ( u, r ( u )) with theequivalent disk D H ( v, r ( u ) − G is isometric in H and is Helly, the corresponding disksin G have a common intersection. Assume now that r ( u ) = 1. Then, v ∈ D H ( u, r ( u )). As thedisks of F pairwise intersect, every w ∈ M \ { u } satisfies r ( w ) + r ( u ) ≥ d H ( w, u ) = d H ( w, v ) + 1 = d H ( w, v ) + r ( u ). Hence, d H ( w, v ) ≤ r ( w ) and vertex v is common to all disks. Lemma 13.
Let G + { u } be a graph obtained by adding a vertex u as a true twin to v ∈ V ( G ) . If G is Helly, then G + { u } is Helly.Proof. Let H := G + { u } , F = (cid:8) D H ( w, r ( w )) : w ∈ M ⊆ V ( H ) (cid:9) be a family of pairwise intersectingdisks in H , and u ∈ M . Because u is a true twin of v , D H ( u, r ) = D H ( v, r ) for any radius r ≥ F is centered at a vertex of G , therefore there is a commonintersection of all disks of F . Lemma 14.
Let G + { u } be the graph obtained by adding a vertex u as a false twin to v ∈ V ( G ) .If G is Helly and there is some y ∈ V ( G ) with N [ v ] ⊆ N [ y ] , then G + { u } is Helly.Proof. Let H := G + { u } , F = (cid:8) D H ( w, r ( w )) : w ∈ M ⊆ V ( H ) (cid:9) be a family of pairwise intersectingdisks in H , and u ∈ M . If r ( u ) >
1, then D H ( v, r ( u )) = D H ( u, r ( u )) and so we can assume thateach disk of F is centered at a vertex of G , implying a non-empty common intersection of all disksof F . Assume now that r ( u ) = 1. As each disk of F ′ = (cid:0) F \ { N [ u ] } (cid:1) ∪ { N [ v ] } is centered at a14 X u vS yY (a) Data structure before adding w . xX u v wS yY (b) Data structure after adding true twin w of v . xX uS yYw S w vS v (c) Data structure after adding w pendant to v . xX uS yYw S w vS v (d) Data structure after adding false twin w of v . Figure 7: Modification of the data structure when adding a vertex w to G i .vertex of G , there is a common intersection R of all disks of F ′ . Since N [ v ] ∈ F ′ , R ⊆ N [ v ]. Recallthat there is a vertex y in G with N [ v ] ⊆ N [ y ]. Therefore, v ∈ R implies y ∈ R . Thus, there existsvertex s ∈ R ∩ N ( v ). By definition of u , N ( u ) = N ( v ). Thus, s ∈ N ( u ) and hence s is containedin each disk in F . The idea for our data structure is based on the partition refinement data structure. It was in-troduced in [34] and allows to find all twins in a graph in linear time. Similar to a partitionrefinement, our data structure handles sets of vertices. In addition to that, it also adds directededges between sets. We create these sets and edges in such a way that the following two propertiesare satisfied: two vertices are in the same set if and only if they are true twins, and there is an edgefrom a set X to a set Y if and only if N [ x ] ⊂ N [ y ] for each x ∈ X and y ∈ Y .To construct our data structure, we use a pruning sequence ( v , . . . , v n ) of a given graph G .Let G i denote the graph induced by { v , . . . , v i } and let N i [ v ] denote the closed neighborhood of avertex v with respect to G i . For G , our data structure only contains a single set S = { v } . Eachtime we add a vertex to G i , we update our data structure as follows to ensure both properties arestill satisfied.Assume that we have three sets S , X , and Y with u, v ∈ S , x ∈ X , and y ∈ Y . Additionally,let there be an edge from X to S and from S to Y . Hence, N i [ x ] ⊂ N i [ u ] = N i [ v ] ⊂ N i [ y ]. Let G i +1 be the graph created by adding a vertex w . We now have three cases: (i) if w is pendant to v , then create two new sets S v := { v } and S w := { w } , set S := S \ { v } , and add the edges XS v , SS v , and S w S v ; (ii) if w is a true twin of v , set S := S ∪ { w } ; and (iii) if w is false twin of v , createtwo new sets S v := { v } and S w := { w } , set S := S \ { v } , and add the edges S v S , S v Y , S w S , and S w Y . Note that, X and Y are not necessarily unique. Hence, when adding an edge from X or to Y , we have to add such an edge for each such set X and Y . See Figure 7 for an illustration.There is a special case for the second vertex v which only happens for that vertex. Afteradding v , it is a true twin and a pendent vertex to v . The construction of our data structureabove, however, assumes that, if w is pendent to v , then v has some neighbor not adjacent to w .Therefore, to construct the data structure correctly, v should be treated as true twin of v andnot as pendant vertex. Lemma 15.
In the data structure constructed above, two vertices are in the same set if and onlyif they are true twins. To do so, start with the set V ( G ) and, for each vertex v , call Refine( N [ v ]) for true twins or Refine( N ( v )) for falsetwins. roof. The lemma is clearly satisfied for G . Assume now, by induction, that our data structuresatisfies Lemma 15 for G i and let A = S be a set handled by the data structure. After adding w ,either all vertices in A are adjacent to w (if w is a twin of v and vertices in A are adjacent to v ) ornon of them are. All other neighbors remain the same. Hence, all sets A = S still satisfy Lemma 15after adding w . To analyse S , we need to distinguish between the three cases of w . Case (i): w is pendant to v . After adding w , we have N i +1 [ u ] = N i [ u ] for each u ∈ S , N i +1 [ v ] = N i [ v ] ∪ { w } and N i +1 [ w ] = { v, w } . It follows that v and w have no true twins in G i +1 and all remaining vertices in S (with respect to G i +1 ) are still true twins. Hence, by placing v and w into their own respective sets, the data structure still satisfies Lemma 15. Case (ii): w is a true twin of v . In this case, clearly, N i +1 [ u ] = N i +1 [ v ] = N i +1 [ w ] for each u ∈ S . Hence, by adding w to S , the data structure still satisfies Lemma 15. Case (iii): w is a false twin of v . After adding w , we have N i +1 [ u ] = N i [ u ] ∪ { w } for each u ∈ S , N i +1 [ v ] = N i [ v ], and N i +1 [ w ] = N i ( v ) ∪ { w } . It follows that v and w have no true twins in G i +1 and all remaining vertices in S (with respect to G i +1 ) are still true twins. Hence, by placing v and w into their own respective sets, the data structure still satisfies Lemma 15. Lemma 16.
In the data structure constructed above, there is an edge from a set A to a set B ifand only if N [ a ] ⊂ N [ b ] for each a ∈ A and b ∈ B .Proof. The lemma is clearly satisfied for G . Assume now, by induction, that our data structuresatisfies Lemma 16 for G i and let a and b be two vertices in G i with N i [ a ] * N i [ b ]. Clearly, sincewe only add a new vertex and do not remove any existing vertices, N i +1 [ a ] * N i +1 [ b ]. Now assumethat a, b / ∈ { v, w } and N i [ a ] ⊆ N i [ b ]. If a is adjacent to w in G i +1 , then a is adjacent to v in G i . Itfollows that b is adjacent to v and, hence, w too. Therefore, N i [ a ] ⊆ N i [ b ] implies N i +1 [ a ] ⊆ N i +1 [ b ]for all a, b / ∈ { v, w } . Lemma 16 is therefore satisfied for each pair of sets A, B = S . To analyse theedges of S , S v , and S w , we need to distinguish between the three cases of w . Case (i): w is pendant to v . After adding w , we have N i +1 [ a ] = N i [ a ] for each a / ∈ { v, w } , N i +1 [ v ] = N i [ v ] ∪ { w } and N i +1 [ w ] = { v, w } . It follows that the added edges XS v , SS v , and S w S v are needed to satisfy Lemma 16, and that adding any other edge would violate Lemma 16. Case (ii): w is a true twin of v . In this case, clearly, N i +1 [ u ] = N i +1 [ v ] = N i +1 [ w ] for each u ∈ S . Additionally, for each vertex a , N i [ a ] ⊆ N i [ v ] if and only if N i +1 [ a ] ⊆ N i +1 [ v ], and N i [ v ] ⊆ N i [ a ] if and only if N i +1 [ v ] ⊆ N i +1 [ a ]. Hence, the data structure still satisfies the lemmaafter adding w into S . Case (iii): w is a false twin of v . After adding w , we have N i +1 [ u ] = N i [ u ] ∪ { w } for each u ∈ S , N i +1 [ v ] = N i [ v ], N i +1 [ w ] = N i ( v ) ∪ { w } , N i +1 [ x ] = N i [ x ] ∪ { w } , and N i +1 [ y ] = N i [ y ] ∪ { w } .It follows that the added edges S v S , S v Y , S w S and S w Y are needed to satisfy Lemma 16, and thatadding any other edge would violate Lemma 16.Before discussing the efficiency of our data structure, observe the following. If w is a pendantvertex or false twin of v , we remove v from S . It can therefore happen that S becomes empty. Inthat case, instead of removing v from S and creating a new set S v , we leave v in S , S becomes S v ,and we update the edges accordingly. That is, we remove all outgoing edges if w is pendant to v ,or we remove all incoming edges if w is a false twin of v . Lemma 17.
For a given distance-hereditary graph G and a corresponding pruning sequence, theoverall runtime to construct the data structure as described above is at most linear with respect tothe size of G .Proof. When constructing the data structure, we only add edges if a new set is created, an edgebetween two sets is created and removed at most once, and there is at least one edge between twovertices in G for each edge between two sets. Additionally, each set is created at most once and16ontains at least one vertex. Therefore, the overall size of the data structure is at most as large asthe size of G and the runtime to construct it is at most linear. We next show that one can efficiently compute the injective hull of a distance-hereditarygraph G . Moreover, as a byproduct, we get that H ( G ) is distance-hereditary and | V ( H ( G )) | ∈ O ( | V ( G ) | ). One attempt to compute H ( G ) is to add Helly vertices suspending all maximal 2-setsof G . We observe that G has O ( n ) maximal 2-sets. Indeed, since G is chordal [1], there are O ( n ) maximal cliques in G obtainable via a perfect elimination ordering of G , and there is one-to-one correspondence between maximal cliques of G and maximal 2-set of G . Since adding aHelly vertex h to suspend a single 2-set in G may create another unsuspended 2-set in G + { h } ,this information alone is insufficient to conclude but gave a promising indication that possibly | V ( H ( G )) | ∈ O ( | V ( G ) | ). Second attempt based on Lemma 11 is to suspend all extended squares,which incurs a similar problem that G + { h } may have a new unsuspended extended square. Ad-ditionally, there can be more extended squares than there are maximal 2-sets.Using Lemma 12, Lemma 13, Lemma 14, a pruning sequence of a distance-hereditary graph G ,and the data structure described in the previous subsection, we can compute H ( G ) in linear time.Moreover, as a byproduct, we get that H ( G ) is distance-hereditary, too. Theorem 5. If G is a distance-hereditary graph, then H ( G ) is distance-hereditary and can becomputed in O ( n + m ) time, where n = | V ( G ) | and m = | E ( G ) | .Proof. We use a pruning sequence σ G = ( v , . . . , v n ) of G . Let G i denote the graph induced by { v , . . . , v i } and let H be a graph that initially contains only v ; clearly H is Helly, distance-hereditary, and contains G as an isometric subgraph. We iterate over the remaining vertices v i ∈ σ G , i ≥
2, to carefully attach new vertices to H as pendants/twins to old vertices in H ,thereby constructing for H a pruning sequence σ H . Then, by Proposition 7(iii), H is distance-hereditary. Additionally, we maintain a data structure for H as described in Section 7.2 above. Forclarity, denote by H k the graph induced by { u ∈ V ( H ) : σ H ( u ) ≤ k } . We claim that the resultinggraph H k = H ( G ), where k = | V ( H ) | . There are two cases. Case 1. Next vertex v i ∈ σ G is a pendant or true twin to some vertex v j in G i . Set H := H + { v i } (the graph obtained by adding v i as a pendant or true twin to v j in H ). By Lemma 12, andLemma 13, H remains Helly. As H is distance-hereditary and contains G i as an induced subgraph, G i is isometric in H . Case 2. Next vertex v i ∈ σ G is a false twin to some vertex v j in G i . Use the data structurefor H to determine if H contains a vertex y = v j with N [ v j ] ⊆ N [ y ] as follows. Let S be the setcontaining v j . By Lemma 15 and Lemma 16, such a y exits if and only if | S | > S has anoutgoing edge. If H contains no such y , then we first create a new true twin y of v j in H and set H := H + { y } . Next, set H := H + { v i } (the graph obtained by adding v i as a false twin to v j in H ). By Lemma 14 H remains Helly. As H is distance-hereditary and contains G i as an inducedsubgraph, G i is isometric in H .A pruning sequence for G can be constructed in linear time [2, 14, 15]. By Lemma 17, the datastructure for H can be constructed in linear time, too. Checking if H contains a vertex y = v j with N [ v j ] ⊆ N [ y ] (case 2) can then be done in constant time. Note that, for each vertex v i ∈ V ( G ),there is at most one vertex y i added to H . Similarly, for each edge v i v j where v i is a false twin to u in G i , we add at most three additional edges ( y i v i , y i v j , and y i u ) to H . Therefore, H has at most2 n vertices and 4 m edges and can be constructed in O ( n + m ) time.We finally claim that H is a minimal Helly graph that contains G as an isometric subgraph, i.e., H = H ( G ). By contradiction, assume there is vertex y ∈ V ( H ) \ V ( G ) with minimal σ H ( y ) suchthat H \{ y } is Helly. Since y / ∈ V ( G ), by algorithm construction, there is a vertex v i ∈ V ( G i ) which17s a false twin to v j in G i , but no vertex in V ( H σ H ( v i ) ) \{ y } suspends the 2-set M = D G i ( v j , ∪{ v i } in H σ H ( v i ) \ { y } . Since H \ { y } is Helly, there is a vertex u with minimal σ H ( u ) such that u suspends M in H \ { y } , where σ H ( u ) > σ H ( y ). Each v ∈ M has σ H ( v ) < σ H ( u ). Let u be a pendant/twinto vertex z in the graph H σ H ( u ) , where σ H ( z ) < σ H ( u ). Since M ⊆ D H σH ( u ) ( u,
1) and | M | ≥ u is not pendant to z . Hence, u is a twin to z and therefore, M ⊆ D H σH ( u ) ( z, z suspends M , a contradiction with the minimality of σ H ( u ). We show that several restrictive graph classes, including split graphs, cocomparability graphs, bi-partite graphs, and consequently graphs of bounded hyperbolicity, graphs of bounded chordality,graphs of bounded tree-length or tree-breadth, and graphs of bounded diameter can have injec-tive hulls that are exponential in size. In particular, there is a graph G of that class such that | V ( H ( G )) | ∈ Ω( a n ) for some constant a > n = | V ( G ) | .We will use the following lemma to obtain a lower bound on the number of vertices in theinjective hull of a particular graph. Recall that a set S ⊆ V ( G ) is said to be a if all verticesof S have pairwise distance at most 2. Lemma 18. If G has at least k unsuspended maximal 2-sets, then | V ( H ( G )) \ V ( G ) | ≥ k .Proof. Each unsuspended maximal 2-set S = { v , . . . , v ℓ } corresponds to a unique family of pairwiseintersecting disks { D ( v i ,
1) : v i ∈ S } that have no common intersection in G . As H ( G ) is thesmallest Helly graph into which G isometrically embeds, then for each S there is a unique Hellyvertex h ∈ V ( H ( G )) universal to maximal 2-set S in H ( G ), i.e., h ( v i ) = 1 for each v i ∈ S and h ( x ) = d G ( x, S ) + 1 for each x ∈ V ( G ) \ S (see Section 3). A graph is a split graph if there is a partition of its vertices into a clique and an indepen-dent set [29, 36]. We construct a special split graph G as follows. Let X = ( x , x , . . . , x k )be an independent set and let Y = ( y , y , . . . , y k ) be an independent set. Let also M =( u , v , w , z , u , v , w , z , . . . , u k , v k , w k , z k ) be a clique partitioned into k complete graphs K .For each integer i ∈ [1 , k ], let x i be adjacent to u i and v i , and let y i be adjacent to w i and z i .Additionally, for all distinct integers i, j ∈ [1 , k ], let x i be adjacent to u j and z j , and let y i beadjacent to w j and v j . See Figure 8 for an illustration. By construction, each vertex x i ∈ X iswithin distance 2 of every vertex in the graph except y i . Every shortest ( x i , y i )-path goes through M , but y i and x i have no common neighbor in M . However, each x i and y j share a common vertex v i . Observe that the resulting graph G has the following distance properties:- ∀ x i ∈ X, ∀ m ∈ M, d G ( x i , m ) ≤ u i ;- ∀ y i ∈ Y, ∀ m ∈ M, d G ( y i , m ) ≤ w i ;- ∀ i, j ∈ [1 , k ] , i = j, d G ( x i , y j ) ≤ z j ;- ∀ i, j ∈ [1 , k ] , i = j, d G ( x i , x j ) = 2 via common neighbor u j ;- ∀ i, j ∈ [1 , k ] , i = j, d G ( y i , y j ) = 2 via common neighbor w j ;- ∀ i ∈ [1 , k ] , d G ( x i , y i ) = 3 because y i and x i have no common neighbor in M . Theorem 6.
There is a split graph G such that | V ( H ( G )) | ≥ n/ + 2 n/ − , where n = | V ( G ) | .Proof. Clearly, G is a split graph with independent set X ∪ Y and clique M .We first claim that G described above has 2 k maximal 2-sets, where k = n/
6. Let S be amaximal 2-set in G . Since all vertices are within distance at most 2 from M , then M ⊂ S . Itremains only to observe that for each i ∈ [1 , k ], either x i ∈ S or y i ∈ S , but not both since d G ( x i , y i ) = 3. 18 x x k X u u u k v v v k z z z k w w w k M y y y k Y . . .. . .. . .. . .
Figure 8: G is a split graph that requires exponentially many new Helly vertices. For readability,some edges are not shown. X and Y are independent sets and M is a clique of k complete graphs K .We next claim that any maximal 2-set S that contains at least two vertices from X and at leasttwo vertices from Y is unsuspended. By contradiction, suppose a vertex m ∈ V ( G ) suspends S .As X and Y are independent sets, necessarily m ∈ M . Thus, m ∈ { u i , w i , v i , z i } for some i ∈ [1 , k ].However, for all j ∈ [1 , k ], d G ( u i , y j ) = 2 and d G ( w i , x j ) = 2 holds. Hence, m = u i and m = w i .As there are at least two vertices of X in S , there is an x j ∈ S such that d G ( v i , x j ) = 2. As thereare at least two vertices of Y in S , there is a y j ∈ S such that d G ( z i , y j ) = 2. Thus, m = v i and m = z i , a contradiction with the choice of m .Moreover, there are at least 2 k − k − G . Observe that only 2maximal 2-sets S have no x i ∈ S or have no y i ∈ S . There are k maximal 2-sets which have only one x i ∈ S (one i ∈ [1 , k ] is reserved for x i ∈ S and all other j ∈ [1 , k ], j = i , have y i ∈ S ). Similarly,there are k maximal 2-sets which have only one y i ∈ S . By Lemma 18, | V ( H ( G )) \ V ( G ) | ≥ k − k −
2. Including the 6 k vertices of V ( G ), one obtains | V ( H ( G )) | ≥ k + 4 k − G also has tree-length tl ( G ) ≤ tb ( G ) ≤ Corollary 4.
Split graphs, chordal graphs, α -metric graphs, 1-hyperbolic graphs, graphs with tl ( G ) ≤ , tb ( G ) ≤ , and graphs with diam ( G ) ≤ can have exponentially large injective hulls.Specifically, there is a graph G of that class with | V ( H ( G )) | ∈ Ω( a n ) , where a > and n = | V ( G ) | . Cocomparability graphs are exactly the graphs which admit a cocomparability ordering [13], i.e.,an ordering σ = [ v , v , . . . , v n ] of its vertices such that if σ ( x ) < σ ( y ) < σ ( z ) and xz ∈ E ( G ), then xy ∈ E ( G ) or yz ∈ E ( G ) must hold. Cocomparability graphs form a subclass of AT-free graphs.A special cocomparability graph G is constructed as follows. Let X = ( x , x , . . . , x k ) be a cliqueand Y = ( y , y , . . . , y k ) be a clique. Let also M = ( u , v , w , z , u , v , w , z , . . . , u k , v k , w k , z k ) bea clique partitioned into k complete graphs K . For each integer i ∈ [1 , k ], let x i be adjacent to u i and v i , and let y i be adjacent to w i and z i . Additionally, for all distinct integers i, j ∈ [1 , k ], let x i be adjacent to u j and z j , and let y i be adjacent to w j and v j . See Figure 9 for an illustration.We emphasize that the key difference between graph G described above and the chordal graphconstruction in Figure 8 is that, here, X and Y are cliques.By construction, each vertex x i ∈ X is within distance 2 of every vertex in the graph except y i . Every shortest ( x i , y i )-path goes through M , but y i and x i have no common neighbor in M .However, each x i and y j share a common vertex v i . Observe that the resulting graph G has thefollowing distance properties:- ∀ i, j ∈ [1 , k ] , i = j, d G ( x i , x j ) = d G ( y i , y j ) = 1;19 ∀ x i ∈ X, ∀ m ∈ M, d G ( x i , m ) ≤ u i ;- ∀ y i ∈ Y, ∀ m ∈ M, d G ( y i , m ) ≤ w i ;- ∀ i, j ∈ [1 , k ] , i = j, d G ( x i , y j ) ≤ z j ;- ∀ i ∈ [1 , k ] , d G ( x i , y i ) = 3 because y i and x i have no common neighbor. x x x k X u u u k v v v k z z z k w w w k M y y y k Y . . .. . .. . .. . .
Figure 9: G is a cocomparability graph that requires exponentially many new Helly vertices. Forreadability, some edges are not shown. X and Y are each cliques and M is a clique of k completegraphs K . Theorem 7.
There is a cocomparability graph G such that | V ( H ( G )) | ≥ n/ + 2 n/ − , where n = | V ( G ) | .Proof. The proof that G has an exponential number of maximal unsuspended 2-sets is the same asin the proof of Theorem 6, establishing | V ( H ( G )) | ≥ n/ + 2 n/ − G is a cocomparability graph. Let m , m be two vertices of M .Let σ be a vertex ordering of G such that σ ( x ) < σ ( m ) for all x ∈ X , σ ( m ) ≤ σ ( m ) ≤ σ ( m )for all m ∈ M , and σ ( m ) < σ ( y ) for all y ∈ Y . That is, σ is an ordering which consists of allvertices of X , followed by all vertices of M , followed by all vertices of Y . We claim that σ is acocomparability ordering. Since X is a clique, for any x i x j ∈ E ( G ) and any x ∈ X such that σ ( x i ) < σ ( x ) < σ ( x j ) has an edge to x i and x j . We apply the same argument to vertices of cliques M and Y . Thus, any ordering of the vertices of X alone is a cocomparability ordering, any orderingof the vertices of M alone is a cocomparability ordering, and any ordering of the vertices of Y aloneis a cocomparability ordering.We next show that any other possible edges between the sets X, M, Y satisfy the constraintsof a cocomparability ordering. Consider any vertices x ∈ X , m ∈ M , and v ∈ V ( G ) with σ ( x ) <σ ( v ) < σ ( m ) and xm ∈ E ( G ). Then, either v ∈ M and therefore vm ∈ E ( G ), or v ∈ X andtherefore vx ∈ E ( G ). By symmetry, any m ∈ M , v ∈ V ( G ), and y ∈ Y with σ ( m ) < σ ( v ) < σ ( y )and xm ∈ E ( G ) satisfies that either vm ∈ E ( G ) or vy ∈ E ( G ). By construction, all vertices x ∈ X and all y ∈ Y satisfy xy / ∈ E ( G ). Therefore, σ is a cocomparability ordering. Corollary 5.
Cocomparability graphs and AT-free graphs can have exponentially large injectivehulls. Specifically, there is a graph G of that class with | V ( H ( G )) | ∈ Ω( a n ) , where a > and n = | V ( G ) | . Currently, we do not know whether there is a permutation graph G with exponentially largeinjective hull. We construct a special bipartite graph G with 2 k ( k ≥
3) vertices as follows. Let X = { x , x , . . . , x k } be an independent set and let Y = { y , y , . . . , y k } be an independent set. Foreach i, j ∈ [1 , k ] and i = j , let x i y j ∈ E ( G ). See Figure 10 for an illustration. Clearly, no two20ertices in X are adjacent and no two vertices in Y are adjacent. By construction, G has thefollowing distance properties:- ∀ i, j ∈ [1 , k ] , i = j, d G ( x i , y j ) = 1;- ∀ i, j ∈ [1 , k ] , i = j, d G ( x i , x j ) = 2 via common neighbor y p , p = i, j ;- ∀ i, j ∈ [1 , k ] , i = j, d G ( y i , y j ) = 2 via common neighbor x p , p = i, j ;- ∀ i ∈ [1 , k ] , d G ( x i , y i ) = 3 as any x i is adjacent to only vertices y j ∈ Y , j = i . x x x k X y y y k Y . . .. . .
Figure 10: G is a bipartite graph that requires exponentially many new Helly vertices. Non-edgesare drawn in dashed lines. X and Y are independent sets. Theorem 8.
There is a bipartite graph G with no induced C k , k > , such that | V ( H ( G )) | ≥ n/ − , where n = | V ( G ) | .Proof. Clearly, G as constructed above is bipartite and has no induced C k for k >
6. Next, we showthat G has exponentially many unsuspended maximal 2-sets. Observe that there are 2 k maximal2-sets in G that are suspended or unsuspended; for each j ∈ [1 , k ], either x j ∈ S or y j ∈ S , but notboth since d G ( x j , y j ) = 3. We claim that any maximal 2-set S that contains at least two verticesfrom X and at least two vertices from Y is unsuspended. Let x i , x j , y k , y ℓ ∈ S , where i, j, k, ℓ arepairwise distinct integers, x i , x j ∈ X , and y k , y ℓ ∈ Y . By construction, x i , x j , y k , y ℓ induce a C .As X and Y are independent sets, there is no vertex of G that suspends this C and hence S . Asthere are 2 k + 2 maximal 2-sets which do not contain at least two vertices from X and at least twovertices from Y , by Lemma 18, | V ( H ( G )) \ V ( G ) | ≥ k − k −
2. Including the 2 k vertices of V ( G ),one obtains | V ( H ( G )) | ≥ k −
2, where k = n/ Corollary 6.
Bipartite graphs can have exponentially large injective hulls. Specifically, there is agraph G of that class with | V ( H ( G )) | ∈ Ω( a n ) , where a > and n = | V ( G ) | . We proved that chordal graphs, square chordal graphs, and distance-hereditary graphs are closedunder Hellification; permutation graphs are not. We provided a linear-time algorithm to compute H ( G ) when G is distance-hereditary. Additional graph classes are identified for which H ( G ) isimpossible to compute in subexponential time, including split graphs, cocomparability graphs, AT-free graphs, bipartite graphs, and graphs with a constant bound on any of the following parameters:diameter, hyperbolicity, tree-length, tree-breadth, or chordality. Recall that the chordality of agraph G is the size of its largest induced cycle; chordal graphs are exactly the graphs of chordality3. A few interesting questions remain open. As distance-hereditary graphs are square-chordal,can the injective hull of square-chordal graphs be constructed efficiently? Can the injective hullof permutation graphs be constructed efficiently? Are cocomparability graphs or AT-free graphsclosed under Hellification? 21 eferences [1] Hans-J¨urgen Bandelt, Anja Henkmann, and Falk Nicolai. Powers of distance-hereditary graphs. Discrete Mathematics , 145(1):37 – 60, 1995.[2] Hans-J¨urgen Bandelt and Henry Martyn Mulder. Distance-hereditary graphs.
J. Comb. TheorySer. B , 41(2):182–208, October 1986.[3] Hans-J¨urgen Bandelt and Henry Martyn Mulder. Pseudo-modular graphs.
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