Equivalence between Hypergraph Convexities

Let G be a connected graph on V . A subset X of V is all-paths convex or ap-convex if X contains each vertex on every path joining two vertices inX and is monophonically convex orm-convex if X contains each vertex on every chordless path joining two vertices inX. First of all, we prove that ap-convexity and m-convexity coincide in G if and only if G is a tree. Next, in order to generalize this result to a connected hypergraph H, in addition to the hypergraph versions of ap-convexity and m-convexity, we consider canonical convexity or c-convexity and simple-path convexity or sp-convexity for which it is well known that m-convexity is finer than both c-convexity and sp-convexity and sp-convexity is finer than ap-convexity. After proving sp-convexity is coarser than c-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of γ-acyclic hypergraphs.


Introduction
Convexity is a fundamental concept occurring in geometry, topology, and functional analysis, and the problem of computing convex hulls is at the core of many computer engineering applications, for instance, in robotics, computer graphics, or optimization see page 125 in 1 .For the theory of abstract convex structures see 2 .This paper is motivated by the paper by Farber and Jamison 3 where structural properties of different notions of convexity in graphs and hypergraphs are stated.We focus on the following four notions of convexities in hypergraphs: m-convexity for monophonic convexity 4 , c-convexity for canonical convexity 4 , sp-convexity for simple-path convexity 3 , and ap-convexity for all-paths convexity , which are known to be related to each other by the following implications: distinct vertices in X are adjacent.The subgraph of G induced by a nonempty subset X of V G is the graph, denoted by G X , with vertex set X in which two distinct vertices are adjacent if and only if they are adjacent in G.The notation G V \ X is abridged into G \ X.
A path is a sequence p a 0 , a 1 , . . ., a k , k ≥ 1, of distinct vertices such that a i−1 and a i are adjacent for 1 ≤ i ≤ k.The path p is said to join a 0 and a k or, equivalently, to be an a 0 -a k path and to have length k; moreover, if k > 1, p is said to pass through each a h , 1 ≤ h ≤ k − 1, and two vertices a i and a j on p are said to be consecutive if |i − j| 1.By V p we denote the vertex set {a 0 , a 1 , . . ., a k }.
The distance between two vertices u and v is the length of any minimum-length u-v path.A graph G is distance hereditary if, for every two vertices u and v of G and for every connected, induced subgraph G containing u and v, the distances between u and v in G and G are the same.
A cycle of length k, k ≥ 2, is a sequence c a 0 , a 1 , . . ., a k−1 , a 0 where a 0 , a 1 , . . ., a k−1 is a path, and the vertices a 0 and a k−1 are adjacent.Two vertices a i and a j on the cycle c are consecutive if either |i − j| 1 or |i − j| k − 1.By V c we denote the set of vertices {a 0 , a 1 , . . ., a k−1 }.A chord of c is an edge joining two nonconsecutive vertices on c.
A graph is chordal if every cycle of length at least 4 has a chord.A graph is strongly chordal if it is chordal and, for every cycle c of even length, there are two nonconsecutive vertices at odd distance on c that are adjacent.A graph is Ptolemaic if it is distance hereditary and chordal.
Let G be a graph with at least two vertices.A vertex u of G is a cut vertex or an "articulation point" of G if the number of connected components of G \ {u} is greater than the number of connected components of G or, equivalently, there exist two vertices v / u and w / u in the connected component of G containing u such that every v-w path passes through u 14 .A block of G is a maximal connected partial graph of G containing no cut vertices.A block of G is trivial if it consists of two vertices and nontrivial otherwise.Finally, G is a block graph if the vertex set of every block of G is a clique.Proposition 2.1 see 14 .Let G be a nontrivial block.For every three distinct vertices u, v, and w of G, there exists a v-w path that passes through u.Proposition 2.2 see 14 .Let G be a connected graph, and let B be a block of G.If w is not a vertex of B, then B contains a cut vertex u of G such that, for every vertex v / u of B, every v-w path passes through u.
Let G be a graph.Two connected vertices are separated by a subset S of V G if they are in two distinct components of the induced graph G \ X.A nonempty subset X of V G is nonseparable if G X is connected and no two vertices in X are separated by a clique of G.The prime components of G are the subgraphs of G induced by maximal nonseparable sets.

Hypergraphs
Generalizing notions and convexities from graphs to hypergraphs is not always straightforward, because there are often several nonequivalent ways to do this and different terminologies.This is true also for notions that hypergraph convexities are based on.For example, "simple paths" in 3 are called "chordless chains" in 15 , "simple circuits" in 3 are called "chordless cycles" in 15 , and "weak β-cycles" in 3 if they are of length at least 3, "nest" vertices in 3 are called "simple" vertices in 15 .
The following basic definitions are taken from 16 .
A generic hypergraph is a possibly empty set H of nonempty sets; the elements of H are the hyper edges of H and their union is the vertex set of H, denoted by V H .
A hypergraph is trivial if it has only one edge and nontrivial otherwise.A partial sub hypergraph of hypergraph H is any subset of H.A hypergraph is simple if no edge is contained in another edge.The reduction of a hypergraph H is the partial hypergraph of H whose edges are the maximal with respect to set-inclusion edges of H.
Let X be a nonempty subset of V H .The subhypergraph of Hinduced by X is the hypergraph, denoted by H X , whose edges are exactly the maximal with respect to setinclusion edges of the hypergraph {A ∩ X : A partial edge is a nonempty vertex set that is contained in some edge.Two vertices are adjacent if they belong together in some edge.A nonempty subset X of V H is a clique if every two distinct vertices in X are adjacent.A hypergraph is conformal if every clique is a partial edge.
A path is a sequence p a 0 , A 1 , a 1 , . . ., A k , a k , k ≥ 1, where the a i 's are pairwise distinct vertices, the A i 's are pairwise distinct edges, and {a i−1 , a i } ⊆ A i for 1 ≤ i ≤ k.The path p is said to join a 0 and a k or, equivalently, to be an a 0 -a k path , to have length k and, if k > 1, to pass through each a h , 1 ≤ h ≤ k − 1.Moreover, two vertices a i and a j on the path p are consecutive if |i − j| 1.Finally, by V p we denote the vertex set {a 0 , a 1 , . . ., a k }, and by H p we denote the partial hypergraph {A A hypergraph is connected if any two vertices are joined by a path.A nonempty subset The connected components of H are the subhypergraphs of H induced by maximal connected subsets of V H .
The following definitions of an "articulation set" and of a "block" in a hypergraph are the natural generalizations of the notions of a cut vertex and of a block in a graph 8, 17 .
Let H be a reduced hypergraph.A separator is a nonempty subset S of V H such that there exist two connected vertices of H that are in two distinct components of the induced subhypergraph H \ S. A separator of H is an articulation set if it is the intersection of two edges of H.A nonempty subset X of V H is nonseparable if H X is connected and has no articulation set.A block of H is the reduction of the subhypergraph of H induced by a maximal nonseparable set.
{3, 4, 5}, and A 5 {4, 5, 6}.The hypergraph H is shown in Figure 1.The set {4, 5} is the only articulation set of H, and the blocks of H are shown in Figure 2.
Finally, with a hypergraph H we can associate two graphs: the "2-section" of H and the "incidence graph" of H, which are defined as follows.
The 2-section also called "adjacency graph" or "underlying graph" or "primal graph", or "Gaifman graph" of H is the graph with vertex set V H in which two vertices are adjacent if and only if they are adjacent in H.We denote the 2-section of H by H 2 .
The incidence graph of H is the bipartite graph with bipartition V H , H , where a ∈ V H and A ∈ H are joined by an edge if and only if a ∈ A. We denote the incidence graph of H by G H , and the size of H is the number of vertices and edges of G H 18 .Note that, if H is connected, then the size of H is O e where e is the number of edges of G H .

Acyclicity
Fagin 8 introduced four notions of acyclicity for hypergraphs which are now recalled and, in the next sections, will be proven to be closely related to hypergraph convexities.A cycle also called a "circuit" 3 is a sequence c a The cycle c is said to have length k; moreover, two vertices a i and a j on c are consecutive if either |i − j| 1 or |i − j| k − 1.By V c we denote the set of vertices {a 0 , a 1 , . . ., a k−1 }, and by H c we denote the partial hypergraph {A 1 , . . ., A k−1 , A k } of H.
A γ-cycle is a cycle c of length at least 3 such that at most one vertex in V c belongs to three or more edges of H c .
A β-cycle a "weak β-cycle" in 8 is a cycle c of length at least 3 such that every vertex in V c belongs to exactly two edges of H c .
A hypergraph is Berge-acyclic if it contains no cycles, γ-acyclic if it contains no γ-cycles, and β-acyclic if it contains no β-cycles or, equivalently, if every partial hypergraph is αacyclic .
A reduced hypergraph is α-acyclic if all its blocks are trivial hypergraphs.A hypergraph is α-acyclic precisely if its reduction is α-acyclic.
It is well known 8 that the following implications on hypergraphs hold: but none of their reverse implications holds in general.Several characterizations of Berge-acyclicity, γ-acyclicity, β-acyclicity, and α-acyclicity exist, and the following is based on the 2-section of a hypergraph.By Proposition 2.4, β-acyclic hypergraphs are the same as "totally balanced" hypergraphs 3 and as "totally decomposable" hypergraphs in 12 .Finally, note that α-acyclic hypergraphs are called "acyclic" hypergraphs in 4, 9, 11, 17, 18 and "decomposable" hypergraphs in 12, 13, 20, 21 .Before closing this subsection, we mention two α-acyclic hypergraphs which in some sense represent the "superstructure" of a graph and of a hypergraph.
The prime hypergraph of graph G is the reduced hypergraph whose edges are precisely the vertex sets of the prime components of G.

Proposition 2.5 see 21 . The prime hypergraph of a graph is a reduced, α-acyclic hypergraph.
A nonempty subset X of V H is a compact set of H if H X is connected and no partial edge of H X is a separator.Note that if X is a compact set, then H X has no articulation set; but the reverse does not hold in general see Example 2.3 .A compact component of H is the reduction of the subhypergraph of H induced by a maximal compact set.The compact hypergraph of H is the reduced hypergraph whose edges are precisely the vertex sets of the compact components of H.

Convexities in Graphs and Hypergraphs
In this section, we recall the definitions and basic results on some convexities in graphs and hypergraphs.Moreover, we state some preliminary results.
Let H be a connected hypergraph.A set θ H of subsets of V H is a convexity space 24, 25 if i the empty set, the singletons, and The sets in a convexity space θ H are called the θ-convex sets of H.For a subset X of V H , the θ-convex hull of X is the minimal with respect to set inclusion θ-convex set of H that includes X.

Graph Convexities
In this section, we recall the definitions and basic results on monophonic convexity mconvexity and all-paths convexity ap-convexity on a connected graph 14, 17, 24-26 .Let G be a connected graph.By m G and ap G we denote the m-convexity space and the apconvexity space on G, respectively.
A chord of a path p is an edge of G that joins two nonconsecutive vertices on p.A path is chordless or "induced" or "minimal" if it has no chords.A subset X of V G is m-convex if, for every chordless path p joining two vertices in X, each vertex on p belongs to X.
The following result provides a known characterization of m-convex sets.Let X be a subset of for every two distinct vertices u and v in X, u and v are joined by an X-X path, then u and v are adjacent in G.
Let G be a connected graph with n vertices and m edges.Dourado et al. 25 gave an algorithm for computing the m-convex hull of a subset X of V G which runs in O mn 2 |X| 2 time.A better algorithm was given by Kannan and Changat 27 , which runs in O mn time.
A subset X of V G is ap-convex if, for every path p joining two vertices in X, each vertex on p belongs to X.It is easily seen that if G is a tree, then the ap-convex hull of a subset X of V G can be computed in O m time simply by deleting the leaves of G that are not in X.
We now state a characterization of those graphs G on which ap G m G .

Theorem 3.2. Let G be a connected graph.The equality ap G m G holds if and only if G is a tree.
Proof.Since every chordless path is trivially a path, the inclusion ap G ⊆ m G is obvious.
If G is a tree, then trivially one has ap G m G .Assume that G is not a tree.To prove that ap G / m G , consider any nontrivial block of G, say B. Let u and v be two adjacent vertices of B. The path u, v is the only chordless u-v path in G and, hence, {u, v} is m-convex.On the other hand, {u, v} is not ap-convex since, by Propositions 2.1 and 2.2, its ap-convex hull is V B , which proves that ap G / m G .By Theorem 3.2, if G is a tree, then the m-convex hull of a subset X of V G coincides with the ap-convex hull of X and, hence, can be computed in O m time.

Hypergraph Convexities
Let H be a connected hypergraph.In this section we recall the definitions and basic results on monophonic convexity m-convexity , canonical convexity c-convexity , simplepath convexity sp-convexity , and all-paths convexity ap-convexity on H.By m H , c H , sp H , and ap H we denote the m-convexity space, the c-convexity space, the sp-convexity space, and the ap-convexity space on H, respectively.

m-Convexity
Hypergraph m-convexity 4 was not defined in terms of paths but using the hypergraphtheoretic version of Theorem 3.1, that is, a subset X of V H is m-convex if, for every two distinct vertices u and v in X, u and v are joined by an X-X path, then u and v are adjacent in H.
We now recall a useful characterization of m-convex sets.To this end, we need further definitions.
Let X be a subset of V H ; two edges A and B of H are connected outside X, written The edge relation ≡ X is an equivalence relation; the classes of the resultant partition of H will be referred to as the X-components of H.
As we noted above, in contrast with m-convexity in graphs, the original definition of m-convexity in hypergraphs was given without having recourse to any path type.We now prove that m-convexity in hypergraphs can be related to the following generalization of the notion of a chordless path in a graph, which is different from that given in 15 and is defined as follows.
A chord of a path p is a pair of nonconsecutive vertices on p which are adjacent in H.A path in H is chordless if it has no chords.Theorem 3.5.Let H be a connected hypergraph.A subset X of V H is m-convex if and only if, for every chordless path p joining two vertices in X, each vertex on p belongs to X.
Proof. if Assume that X contains V p for every chordless path p joining two vertices in X.By Theorem 3.4, it is sufficient to prove that the boundary of X with every X-component of H is a clique of H. Suppose, by contradiction, that there exists an X-component K of H such that the boundary of X with K contains two vertices u and v that are not adjacent in H. Let Y be the boundary of X with K. Since u and v are not adjacent in H, u and v are not adjacent in K and, since K is X-connected, there exists a u-v path p in K of length at least 2 such that Y ∩ V p {u, v}.Let p be a u-v path of minimum length in K p .Of course, V p ⊆ V p and p is a chordless path; moreover, one has Y ∩V p {u, v}.Since u and v are not adjacent in K, p must be of length at least 2 and, hence, there exists a vertex w in V p ∩ V K that does not belong to Y .Since X ∩ V p Y ∩ V p {u, v}, w does not belong to X so that X does not contain V p contradiction .
only if Assume that X is m-convex.By Theorem 3.4, the boundary of X with every X-component of H is a clique of H. Suppose, by contradiction, that there exist two vertices u and v in X and a chordless u-v path p a ∈ X} and let j min{h > i : a h ∈ X}.Of course, both a i−1 and a j belong to the boundary of X with the X-component of H containing a i and, since the boundary of X with every X-component of H is a clique, a i−1 and a j are adjacent.Since a i−1 and a j are nonconsecutive on p, the pair {a i , a j } is a chord of p contradiction .
Finally, let H be a connected hypergraph with n vertices and m edges.An algorithm for computing m-convex hulls in a hypergraph H is the "monophonic-closure algorithm" 4 which runs in O mn time if the prime hypergraph of H 2 is given.Since the time needed to construct the prime hypergraph of H 2 is O en 21 , where e is the number of edges of H 2 , and since e O n 2 , the time complexity of the monophonic-closure algorithm is O mn n 3 .On the other hand, it is easy to check that a 0 , A 1 , a 1 , . . ., A k , a k is a chordless path in H if and only if a 0 , a 1 , . . ., a k is a chordless path in H 2 , which implies that m H m H 2 so that, given H 2 , the m-convex hull of any vertex set in H can be computed in O ne time, that is, in O n 3 time using the Kannan-Changat algorithm.

c-Convexity
A subset X of V H is c-convex if the boundary of X with every X-component of H is a partial edge of H. Let H be a connected hypergraph with n vertices and m edges.It is proven in 4 that c-convex hulls can be computed using the Maier-Ullman algorithm 28 , which runs in O m 4 n time.A more efficient algorithm is the "canonical-closure algorithm" 4 , which runs in O mn time if the compact hypergraph of H is given.Since the time needed to construct the compact hypergraph of H is O m 3 n 23 , the time complexity of the canonical-closure algorithm is O m 3 n .Note that if H is α-acyclic, then the time complexity of the algorithm reduces to O mn ; however, we can do better using the "selective-reduction algorithm" 18 which is linear in the size of H.

sp-Convexity
A path p in H is simple 3 if every two nonconsecutive vertices on p are not adjacent in the partial hypergraph H p ; equivalently, a path p in H is simple if p is a chordless path in H p .A subset X of V H is sp-convex if, for every simple path p joining two vertices in X, each vertex on p belongs to X. Let H be a connected hypergraph with n vertices and m edges.An efficient algorithm for computing sp-convex hulls was given in 29 , which runs in O mn 3 e , where e is the number of edges of the incidence graph of H. Since e ≤ mn, the time complexity of the algorithm is O m 2 n 4 .However, if H is β-acyclic, then using the Anstee-Farber algorithm 15 , the sp-convex hull of vertex set X can be computed in O mn 2 time simply by deleting the "nest" vertices of H that are not in X see 12 .

ap-Convexity
Let H be a connected hypergraph.The convexity space ap H is defined in the same way as in Section 3.1, that is, a subset X of V H is ap-convex if, for every path p joining two vertices in X, each vertex on p belongs to X. Again one always has ap H ⊆ sp H .In Section 6 we will give an efficient algorithm for computing ap-convex hulls.

c-Convexity versus m-Convexity
Since every partial edge is a clique, one always has c H ⊆ m H .In this section we characterize the class of hypergraphs H for which c H m H . First of all, observe that if H is conformal, then every clique is a partial edge so that, by Theorem 3.4, every m-convex set of H is also c-convex so that c H m H .We will see that conformality is not a necessary condition for c H m H .A clique X of H is a boundary clique if there exists an X-component K of H such that X equals the boundary of X with K.A hypergraph H is weakly conformal if every boundary-clique of H is a partial edge of H.Of course, every conformal hypergraph is weakly conformal.The following is an example of a weakly conformal hypergraph that is not conformal.only if Let X be any boundary clique of H. Since X is a clique, from the very definition of m-convexity it follows that X is m-convex and, since c H m H by hypothesis, X is cconvex.From the very definition of c-convexity, it follows that the boundary of X with every X-component of H is a partial edge.Finally, since X is a boundary clique of H, there exists an X-component K of H such that X is the boundary of X with K. So, X is a partial edge of H.It follows that H is weakly conformal.

sp-Convexity versus c-Convexity and m-Convexity
In this section we characterize the class of hypergraphs H for which sp H c H and the class of hypergraphs H for which sp H m H .

Equivalence between sp-Convexity and c-Convexity
Let H be a connected hypergraph.We first prove that sp H ⊆ c H .To achieve this, we need the following two technical lemmas.
Lemma 5.1.If p is a u-v path in H and u and v are not adjacent in H p , then there exists in H p a simple u-v path p of length at least 2 and with V p ⊆ V p .
Proof.Let a 0 , A 1 , a 1 , . . ., A k , a k be a u-v path.Let i 1 max {h : a h and a 0 are adjacent in H p } and let A j 1 be an edge of H p that contains both a i 1 and a 0 .Since u and v are not adjacent in H p , one has i 1 < k.Consider the u-v path a 0 , A j 1 , a i 1 , . . ., A k , a k .Let i 2 max{h > i 1 : a h and a i 1 are adjacent in H p }, and let A j 1 be any edge of H p that contains both a i 2 and a i 1 .If i 2 k, then the u-v path a 0 , A j 1 , a i 1 , A j 2 , a k is simple since a k / ∈ A j 1 and a 0 / ∈ A j 2 .If i 2 < k, then let i 3 max{h > i 2 : a h and a i 2 are adjacent in H p }, and let A j 3 be any edge of H p that contains both a i 3 are a i 2 .If i 3 k, then the u-v path a 0 , A j 1 , a i 1 , A j 2 , a i 2 , A j 3 , a k is simple since a k / ∈ A j 2 and a i 1 / ∈ A j 3 , and so on.
The next lemma characterizes sp-convex sets.
Lemma 5.2.A subset X of V H is sp-convex if and only if either |X| ≤ 1 or, for every two distinct vertices u and v in X, there exists no X-X path joining u and v in the partial hypergraph H uv of H obtained by deleting the edges that contain both u and v.
Proof.only if Assume that X is sp-convex and |X| > 1. Suppose, by contradiction, that there exist two vertices u and v in X and an X-X path joining u and v in H uv .By construction of H uv , the vertices u and v are not adjacent in H uv so that by Lemma 5.1, there exists a simple X-X path p of length at least 2 joining u and v in H uv .Since p is also a simple path in H and V p is not contained in X, X is not sp-convex contradiction .
if If |X| ≤ 1 then X is trivially sp-convex.Assume that |X| > 1 and, for every two distinct vertices u and v in X, there exists no X-X path joining u and v in H uv .Suppose, by contradiction, that X is not sp-convex.Then, there exist two vertices u and v in X and a simple u-v path p in H with V p \ X / ∅.Let w be a vertex in V p \ X, and let p be the subpath of p such that w is on p and p is an X-X path.Of course, p is a simple path in H and has length at least 2. Let x and y be the vertices in X that are joined by p .Since p is a simple path in H, no edge of H p contains both x and y, so that p is also a path in H xy .To sum up, p is an X-X path that in H xy joins the vertices x and y in X so that X is not sp-convex contradiction .4}, and A 6 {3, 4, 5}.The hypergraph H is shown in Figure 5.Let X {1, 3, 4}.Consider the three vertex pairs in X.For the vertex pair {1, 3}, the partial hypergraph H 13 of H is {A 1 , A 2 , A 4 , A 5 , A 6 } and, since 1 and 3 belong to distinct connected components of H 13 , there exists no X-X path joining 1 and 3 in H 13 .For the vertex pair {1, 4}, the partial Figure 5 hypergraph Proof.Suppose, by contradiction, that there exists an sp-convex set X of H that is not cconvex.Then, there exists an X-component K of H such that the boundary of X with K i.e., the set X ∩ V K is not a partial edge of H.Of course, |X ∩ V K | ≥ 2 and the boundary of X with K is not a partial edge of K. Let A be an edge of K such that for every edge A of K, either X ∩A ⊆ A or there exists u ∈ X ∩A \A.Since the boundary of X with K is not a partial edge of K, X ∩ A is a proper subset of X ∩ V K ; therefore, there exist two vertices u 0 and v such that u 0 ∈ X ∩A and v ∈ X ∩V K \A .Let A be any edge of K that contains v. Since K is an X-component of H, A ≡ X A , and, hence, there exists an X-X path p 0 u 0 , A , v 1 , . . ., A k , v in K of length at least 2 i.e., k ≥ 2 with A k A .If no edge of H p 0 contains both u 0 and v, then X is not sp-convex by Lemma 5.2 and a contradiction arises.Otherwise, let A h 1 be the first edge on p 0 that contains both u 0 and v. Since v ∈ X ∩ A h 1 and v / ∈ A , X ∩ A h 1 is not a subset of A so that there exists If no edge of H p 1 contains both u 1 and v, then X is not sp-convex by Lemma 5.2 and a contradiction arises.Otherwise, let A h 2 be the first edge on p 1 that contains both u 1 and v. Since v ∈ X ∩ A h 2 and v / ∈ A , X ∩ A h 2 is not a subset of A so that there exists u 2 ∈ X ∩ A \ A h 2 .Consider the X-X path p 2 u 2 , A , w 0 , . . ., A h 2 , v .If no edge of H p 2 contains both u 2 and v, then X is not sp-convex by Lemma 5.2 and a contradiction arises, and so on.Thus, ultimately one obtains an X-X path p of length at least 2 that joins two vertices u and v in X and is such that no edge of H p contains both u and v.By Lemma 5.2, X is not sp-convex and a contradiction arises.
We now characterize the class of hypergraphs H for which sp H c H .

Theorem 5.5. Let H be a connected hypergraph. The equality sp H c H holds if and only if H is γ-acyclic.
Proof. if Assume that H is γ-acyclic and suppose, by contradiction, that sp H / c H . Let X be a c-convex set that is not sp-convex.Then, there exist two vertices u and v in X and a simple u-v path p of length at least 2 such that V p \ X / ∅.Let p a 0 , A 1 , a 1 , A 2 , . . ., a k−1 , A k , a k , let h be such that a h / ∈ X, let i max{l : 0 ≤ l < h and a l ∈ X}, and let j min{l : h < l ≤ k and a l ∈ X .Consider the subpath p a i , A i 1 , a i 1 , . . ., A j , a j of p. Since p is a simple path, p is a simple path of length at least 2, and V p ∩ X {a i , a j } by construction.Let K be the X-component of H such that H p ⊆ K.Then, both a i and a j belong to the boundary of X with K. Since X is c-convex, the boundary of X with K is contained in an edge of H, say A. Since p is a simple path of length at least 2, and A contains both a i and a j , A is not an edge of H p so that c a i , A i 1 , a i 1 , . . ., A j , a j , A, a i is a cycle; moreover, since p is a path of length at least 2, c has length at least 3. Distinguish two cases depending on whether or not V p ⊆ A.
A, a i is a cycle of length 3 and, since only the vertex a i 1 belongs to the three edges of c ,c is a γ-cycle contradiction .
Case 2 V p \ A / ∅ .Then, there exists in V p a vertex a r / ∈ A for some r, i < r < j.Let i * max{l : i ≤ l < r and a l ∈ A} and j * min{l : r < l ≤ j and a l ∈ A}.Then c a i * , A i * 1 , a i * 1 , . . ., a j * −1 , A j * , a j * , A, a i * is a cycle of length at least 3 and, since every vertex in V c belongs to exactly two edges of H c , c is a γ-cycle contradiction .
only if Assume that every c-convex set of H is also sp-convex and suppose, by contradiction, that H is not γ-acyclic.Let c a 0 , A 1 , a 1 , A 2 , . . ., a k−1 , A k , a 0 , k ≥ 3, be a γcycle.Distinguish two cases depending on whether or not each vertex in V c belongs to exactly two edges of H c .
is a partial edge of H and, hence, X is c-convex.On the other hand, since each vertex in V c belongs to exactly two edges of H c , a 2 , . . ., A k , a 0 , A 1 , a 1 is simple path of length at least 2 and, since a 0 / ∈ X, X is not sp-convex contradiction .
Case 2. There exists a vertex in V c that belongs to more than two edges of H c .Without loss of generality, let it be a 0 .Since c is a γ-cycle, each a h , h / 0, belongs to exactly two edges of H c .Let A i be any edge of H c \ {A 1 , A k } containing a 0 , and let It is easy to see that a i , A i 1 , a i 1 , . . ., A s , a 0 , A r , a r , . . ., A i−1 , a i−1 is a simple path of length at least 2 and, since a 0 / ∈ X, X is not sp-convex contradiction .
Let H be a connected hypergraph with n vertices and m edges.If H is γ-acyclic, then H is β-acyclic and, hence, sp-convex hulls can be computed in O mn 2 using the Anstee-Farber algorithm.On the other hand, if H is γ-acyclic, then H is α-acyclic and, hence, c-convex hulls can be computed in linear time using the Tarjan-Yannakakis algorithm.By Theorem 5.5, spconvex hulls can be computed in linear time, that is, more efficiently than using the Anstee-Farber algorithm.

Equivalence between sp-Convexity and m-Convexity
Note that, by Theorem 3.5 and by the fact that every chordless path in H is a simple path in H, one always has sp H ⊆ m H .The following is another convexity-theoretic characterization of γ-acyclic hypergraphs.By Theorem 5.6, m-convex hulls can be computed in linear time more efficiently than using the monophonic-closure algorithm.

ap-Convexity versus sp-Convexity, c-Convexity, and m-Convexity
Let H be a connected hypergraph.In this section we characterize the three classes of hypergraphs H for which ap H sp H , ap H c H , and ap H m H .To achieve this, we first give a polynomial algorithm for computing ap-convex hulls.

Computing ap-Convex Hulls
We represent H by its incidence graph G H . Remark 6.1.For every two vertices u and v of H, every u-v path in H is a u-v path in G H and vice versa; moreover, every cycle in H is a cycle in G and vice versa.
To avoid ambiguities, we call the vertices and edges of G H the nodes and arcs of G H , respectively.A node v of G H is a vertex-node or an edge-node depending on whether v ∈ V H or v ∈ H.Moreover, we call cutpoints the cut vertices of G H ; furthermore, a cutpoint of G H is a vertex-cutpoint or an edge-cutpoint depending on whether it is a vertexnode or an edge-node.Note that, if a is a vertex-cutpoint of G H , then either the induced subhypergraph H \ {a} is not connected see the vertex-node 3 in Figure 6 or the singleton {a} is an edge of H see the vertex-node 1 in Figure 6 ; moreover, if A is an edge-cutpoint of G H , then either the partial hypergraph H \ {A} is not connected see the edge-node A 3 in Figure 6 or there exist one or more vertices of H that belong to A and to no other edge of H see the edge-node A 6 in Figure 6 .Our algorithm works with the "block-cutpoint tree" of G H , which is defined as follows.Let T be the bipartite graph whose nodes are the cutpoints and blocks of G H and where v, B is an arc if the cutpoint v of G H is a node of the block B of G H .A node of T is a block-node if it is a block of G H and a cutpoint-node otherwise.It is well known 14 that T is a tree, which is called the block cut-vertex tree of G H .We also label each block-node B of T by the vertex set V B .Example 6.1.Consider again the hypergraph H {A 1 , A 2 , A 3 , A 4 , A 5 , A 6 } of Example 5.3 see Figure 5 .The incidence graph G H of H is shown in Figure 6, in which the cutpoints of G H are circled.
Edge-nodes Vertex-nodes Figure 6 Note that the induced subhypergraph H \ {1} is connected and the induced subhypergraph H \ {3} is not connected; moreover, the partial hypergraph H \ {A 3 } is not connected and the partial hypergraph H \ {A 5 } is connected.
The blocks of G H are reported in Figure 7.
The block-cutpoint tree T of G H is shown in Figure 8.The six block-nodes of T are labeled as follows: block-node label

6.1
Note that each leaf of T is a block-node.Moreover, if H is not a trivial hypergraph and T is a one-point tree, then the node of T is a nontrivial block of G H . Finally, if B is a trivial block of G H with H B {A} and V B {u}, then the block-node B of T is not a leaf if and only if u and A are both cutpoints of G H .
Algorithm 1 constructs the ap-convex hull Y of any subset X of V H . Example 6.1 continued .When we apply Algorithm 1 with input X {1, 3}, the tree T resulting from the pruning of T is shown in Figure 9. So, the output of Algorithm 1 is Y {1, 2, 3}.
When we apply Algorithm 1 with input X {1, 4}, the tree T resulting from the pruning of T is shown in Figure 10.So, the output of Algorithm 1 is Y {1, 2, 3, 4}.7 for, otherwise, the block-node B would be a leaf of T and u would be the only node of T adjacent to B and, since V B ∩ X {u}, the leaf B of T would have been deleted at step 1 , ii if u / ∈ X then B is not a leaf of T see the block-node B 3 in Figure 10 for, otherwise, since V B ∩ X ∅, the block-node B would have been deleted at step 1 .
Fact 3. If a cutpoint-node of T is a vertex-node of G H , say u see the cutpoint-node 3 in Figure 8 , then there exist two leaves B and B of T such that every B-B path in T passes through the node u.Furthermore, since V B ∩ X / ∅ by Fact 1 , there exists a vertex v / u in X ∩ V B for, otherwise, X ∩ V B {u} and, since u is the node adjacent to the leaf B, the leaf B of T would have been deleted at step 1. Analogously, there exists a vertex v / u in X ∩ V B .Finally, every v-v path in G H passes through u.Theorem 6.3.Let H be a connected hypergraph and let X be a subset of V H . Algorithm 1 correctly computes the ap-convex hull of X.
Proof.Let Y be the output of Algorithm 1 and let X ap denote the ap-convex hull of X.If |X| ≤ 1, then Y X see step 2 and X ap X, which proves the statement.Assume that |X| > 1.We first prove that X ap ⊆ Y and, then, Y ⊆ X ap .The following result, which is an immediate consequence of Theorem 6.3, will be used in the sequel.Corollary 6.4.Let H be a connected hypergraph.For every block B of G H , the subsets of V B that are ap-convex are precisely the empty set, the singletons and V B .

Equivalence between ap-Convexity and sp-Convexity
Recall from Section 3.2.4 that one always has ap H ⊆ sp H .We will prove that ap H sp H if and only if the incidence graph G H of H satisfies the following two conditions: C1 every edge-cutpoint of G H is a node of only trivial blocks of G H ; C2 for every nontrivial block B of G H , and for every X ⊆ V B with 1 < |X| < |V B |, there exist two distinct vertex-nodes u and v in X and an X-X path joining u and v in the induced subgraph B uv of B obtained by deleting the edge-nodes that are adjacent to both u and v.
Note that every graph satisfies C1 owing to the fact that every edge contains exactly two vertices.
Remark Proof of (i).Suppose, by contradiction, that condition C1 does not hold.Then, there exists an edge-cutpoint A of G H that is a node of a nontrivial block B of G H . Let v ∈ A ∩ V B , let u ∈ A \ V B , and let B be the block containing both A and u see Figure 13 .By Theorem 6.3, the ap-convex hull of {u, v} is V B V B , which is a proper superset of {u, v} as B is a nontrivial block of G H . On the other hand, since u, A, v is the only simple u-v path in H, the set {u, v} is sp-convex, which contradicts the hypothesis ap H sp H .
Proof of (ii).Let B be any nontrivial block of G H .Note that, since C1 holds, by Remark 6.5 one has |V B | ≥ 3. Let X be any subset of V B with 1 < |X| < |V B |.By Corollary 6.4, the set X is not ap-convex and, since ap H sp H , X is not sp-convex.By Lemma 6.6, there exist two distinct vertex-nodes u and v in X and an X-X path joining u and v in G uv .Since X ⊆ V B , by Proposition 2.2 every path joining u and v in G uv is also a path in B uv ; therefore, there exists an X-X path joining u and v in B uv , which proves that condition C2 holds.
if Let X be any subset of V H with |X| > 1, let X ap and X sp denote the apconvex hull and the sp-convex hull of X, respectively.Of course, X ⊆ X sp ⊆ X ap .Therefore, it is sufficient to prove that X ap ⊆ X sp .By Theorem 6.3, X ap is the output of Algorithm 1, that is, if T is the tree resulting from the pruning of the block-cutpoint tree of G H , then X ap is the union of the sets V B for all block-nodes B of T .So, we need to prove that, for every block-node B of T , one has V B ⊆ X sp .Distinguish two cases depending on whether or not B is a trivial block of G H . Case 1. B is a trivial block of G H . Let u be the unique vertex in V B .If u ∈ X then u ∈ X sp since X ⊆ X sp .Consider now the case that u / ∈ X.Since X ∩ V B ∅, Fact 1, the block-node

Figure 11
Figure 11 4.1.Consider the hyper graph H of Figure 4.The only cliques of H that are not boundary cliques are the two cliques with cardinality 3, namely, the sets {2, 3, 6} and {2, 5, 6}.Since each clique of H with cardinality less than 3 is a partial edge of H, each boundary clique is a partial edge and, hence, H is weakly conformal.Let X be any m-convex set.Let K be any X-component of H, and let Y be the boundary of X with K.By the very definition of m-convexity, Y is a clique; moreover, K is also a Y -component of H and the boundary of Y with K is Y itself.Therefore, Y is a boundary clique of H. Since H is weakly conformal, Y is a partial edge of H.It follows that the boundary of X with every X-component of H is a partial edge of H and, hence, X is c-convex.
14of H is H itself and, since every path joining 1 and 4 in H 14 passes through 3, there exists no X-X path joining 1 and 4 in H 14 .For the vertex pair {3, 4}, the partial hypergraph H 34 of H is {A 1 , A 2 , A 3 , A 4 } and, since 4 is not a vertex of H 14 , there exists no X-X path joining 3 and 4 in H 34 .By Lemma 5.2, the set X is sp-convex, which is confirmed by the fact that the only simple paths joining two vertices in X are 1, A 3 , 3 , 1, A 3 , 3, A 5 , 4 , 1, A 3 , 3, A 6 , 4 , 3, A 5 , 4 , and 3, A 6 , 4 .
We first prove the inclusion sp H ⊆ c H . Theorem 5.4.Let H be a connected hypergraph.Every sp-convex set of H is c-convex.
V B | 1 and the vertex in X ∩ V B is the cutpoint-node adjacent to B. Let T be the resultant tree.3SetY: ∅.For each block-node B of T , set Y : Y V B .Remark 6.2.Each time a block-node B is deleted during the pruning process, either X ∩V B ∅ or |X ∩V B | 1 and the vertex in X ∩V B belongs to V B for some undeleted block-node B .Therefore, one has that X is a subset of Y .Fact 1.Each leaf of T is a block-node and, if B is a leaf of T , then V B ∩ X / ∅.If |X| > 1 andT is a one-point tree, then the block-node of T is a nontrivial block of G H .
{u}.If |X| > 1, then T is not a one-point tree and, furthermore, i if u ∈ X then A is a node of T adjacent to B see the cutpoint-node A 3 and the block-node B 3 in Figure 6.5.If C1 holds then, for every nontrivial block B of G H , one has |V B | ≥ 3 for, otherwise i.e., if |V B | 2 , B would contain an edge-cutpoint of G H see Figure12owing to the fact that distinct edge-nodes of G H have distinct sets of adjacent vertex-nodes.In order to characterize the class of hypergraphs H for which ap H sp H , we first restate Lemma 5.2 as follows.Let G be the incidence graph of H.A subset X of V H is sp-convex if and only if either |X| ≤ 1 or, for every two distinct vertex-nodes u and v in X, there exists no X-X path joining u and v in the induced graph G uv of G H obtained by deleting the edge-nodes adjacent to both u and v. Let G be the incidence graph of H.If condition (C2) holds then, for every block B of G H , the subsets of V B that are sp-convex are precisely the empty set, the singletons and V B .Proof.The empty set and the singletons are trivially sp-convex.Moreover, by Lemma 6.6 and condition C2 , for every nontrivial block of G H , no subset X of V B with 1< |X| < |V B | is sp-convex.Let H be a connected graph.The equality ap H sp H holds if and only if the incidence graph G H of H satisfies both conditions (C1) and (C2).