Abstract
Let be a connected graph on . A subset of is all-paths convex (or -convex) if contains each vertex on every path joining two vertices in and is monophonically convex (or -convex) if contains each vertex on every chordless path joining two vertices in . First of all, we prove that -convexity and -convexity coincide in if and only if is a tree. Next, in order to generalize this result to a connected hypergraph , in addition to the hypergraph versions of -convexity and -convexity, we consider canonical convexity (or -convexity) and simple-path convexity (or -convexity) for which it is well known that -convexity is finer than both -convexity and -convexity and -convexity is finer than -convexity. After proving -convexity is coarser than -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 -acyclic hypergraphs.
1. 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: -convexity (for monophonic convexity) [4], -convexity (for canonical convexity) [4], -convexity (for simple-path convexity) [3], and -convexity (for all-paths convexity), which are known to be related to each other by the following implications: Note that hypergraph -convexity generalizes graph -convexity [3, 5], and both hypergraph -convexity and -convexity generalize graph -convexity [6, 7]. After proving the implication -convex -convex, we characterize the hypergraphs in which each pair of the four convexities above is equivalent.
As an application, suppose that for two hypergraph convexities and we have two algorithms and to compute -convex hulls and -convex hulls for hypergraphs and that the computational complexity of is less than . If and are equivalent in a class of hypergraphs then, for every hypergraph belonging to this class, we can compute -convex hulls using the algorithm instead of .
As a consequence of our equivalence results, we obtain the following convexity-theoretic characterizations of the so-called “Berge-acyclic” hypergraphs and “-acyclic” hypergraphs [8]:(i)a hypergraph is Berge-acyclic if and only if -convexity and -convexity are equivalent in the hypergraph (see Theorem 6.9);(ii)a hypergraph is Berge-acyclic if and only if -convexity and -convexity are equivalent in the hypergraph (see Theorem 6.10);(iii)a hypergraph is -acyclic if and only if -convexity and -convexity are equivalent in the hypergraph (see Theorem 5.5);(iv)a hypergraph is -acyclic if and only if -convexity and -convexity are equivalent in the hypergraph (see Theorem 5.6).
It is worth noting that Berge-acyclic and -acyclic hypergraphs belong to a family of hypergraphs (, and Berge-acyclic hypergraphs) which enjoy a number of theoretical and computational properties in database theory [8, 9], artificial intelligence [10, 11], and statistics [12, 13]. Moreover, -convexity is a “convex geometry” in -acyclic hypergraphs [4], and -acyclicity characterizes those hypergraphs in which -convexity yields a “convex geometry” [3].
To achieve the above-mentioned results, we need several standard hypergraph-theoretic definitions (such as the four acyclicity types above) and additional notions which are introduced ad hoc. The outline of the paper is as follows. Section 2 contains basic definitions and results on graphs and hypergraphs. In Section 3 we review basic results on structural properties of -convexity, -convexity, -convexity, and -convexity in graphs and hypergraphs; moreover, we state some preliminary results. In Section 4 we characterize the hypergraphs in which -convexity and -convexity are equivalent. In Section 5 we prove that -acyclicity characterizes those hypergraphs in which -convexity (or -convexity) and -convexity are equivalent. In Section 6 we characterize the hypergraphs in which -convexity and -convexity are equivalent; moreover, we prove that Berge-acyclicity characterizes those hypergraphs in which -convexity (or -convexity) and -convexity are equivalent. Section 7 contains an open problem for future research.
2. Terminology and Notation
2.1. Graphs
Henceforth, we only consider graphs [14] with no loops and no multiple edges, which henceforth will be referred simply to as graphs. Let be a graph. Two vertices of are adjacent if they are joined by some edge of . A nonempty subset of is a clique if every two distinct vertices in are adjacent. The subgraph of induced by a nonempty subset of is the graph, denoted by , with vertex set in which two distinct vertices are adjacent if and only if they are adjacent in . The notation is abridged into .
A path is a sequence , , of distinct vertices such that and are adjacent for . The path is said to join and (or, equivalently, to be an – path) and to have length ; moreover, if , is said to pass through each , , and two vertices and on are said to be consecutive if . By we denote the vertex set .
The distance between two vertices and is the length of any minimum-length – path. A graph is distance hereditary if, for every two vertices and of and for every connected, induced subgraph containing and , the distances between and in and are the same.
A cycle of length , , is a sequence where is a path, and the vertices and are adjacent. Two vertices and on the cycle are consecutive if either or . By we denote the set of vertices . A chord of is an edge joining two nonconsecutive vertices on .
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 of even length, there are two nonconsecutive vertices at odd distance on that are adjacent. A graph is Ptolemaic if it is distance hereditary and chordal.
Let be a graph with at least two vertices. A vertex of is a cut vertex (or an “articulation point”) of if the number of connected components of is greater than the number of connected components of or, equivalently, there exist two vertices and in the connected component of containing such that every – path passes through [14]. A block of is a maximal connected partial graph of containing no cut vertices. A block of is trivial if it consists of two vertices and nontrivial otherwise. Finally, is a block graph if the vertex set of every block of is a clique.
Proposition 2.1 (see [14]). Let be a nontrivial block. For every three distinct vertices , and of , there exists a – path that passes through .
Proposition 2.2 (see [14]). Let be a connected graph, and let be a block of . If is not a vertex of , then contains a cut vertex of such that, for every vertex of B, every – path passes through .
Let be a graph. Two connected vertices are separated by a subset of if they are in two distinct components of the induced graph . A nonempty subset of is nonseparable if is connected and no two vertices in are separated by a clique of . The prime components of are the subgraphs of induced by maximal nonseparable sets.
2.2. 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 -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 of nonempty sets; the elements of are the (hyper) edges of and their union is the vertex set of , denoted by .
A hypergraph is trivial if it has only one edge and nontrivial otherwise. A partial (sub) hypergraph of hypergraph is any subset of . A hypergraph is simple if no edge is contained in another edge. The reduction of a hypergraph is the partial hypergraph of whose edges are the maximal (with respect to set-inclusion) edges of .
Let be a nonempty subset of . The subhypergraph of induced by is the hypergraph, denoted by , whose edges are exactly the maximal (with respect to set-inclusion) edges of the hypergraph . The notation is abridged into .
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 of is a clique if every two distinct vertices in are adjacent. A hypergraph is conformal if every clique is a partial edge.
A path is a sequence , , where the ’s are pairwise distinct vertices, the ’s are pairwise distinct edges, and for . The path is said to join and (or, equivalently, to be an – path), to have length and, if , to pass through each , . Moreover, two vertices and on the path are consecutive if . Finally, by we denote the vertex set , and by we denote the partial hypergraph of .
A hypergraph is connected if any two vertices are joined by a path. A nonempty subset of is connected if the induced subhypergraph is connected. Note that if is a graph, then the subgraph of induced by equals the reduction of .
The connected components of are the subhypergraphs of induced by maximal connected subsets of .
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 be a reduced hypergraph. A separator is a nonempty subset of such that there exist two connected vertices of that are in two distinct components of the induced subhypergraph . A separator of is an articulation set if it is the intersection of two edges of . A nonempty subset of is nonseparable if is connected and has no articulation set. A block of is the reduction of the subhypergraph of induced by a maximal nonseparable set.
Example 2.3. Let , where , , , , and . The hypergraph is shown in Figure 1. The set is the only articulation set of , and the blocks of are shown in Figure 2.
Finally, with a hypergraph we can associate two graphs: the “2-section” of and the “incidence graph” of , which are defined as follows.
The 2-section (also called “adjacency graph” or “underlying graph” or “primal graph”, or “Gaifman graph”) of is the graph with vertex set in which two vertices are adjacent if and only if they are adjacent in . We denote the 2-section of by .
The incidence graph of is the bipartite graph with bipartition , where and are joined by an edge if and only if . We denote the incidence graph of by , and the size of is the number of vertices and edges of [18]. Note that, if is connected, then the size of is where is the number of edges of .
2.3. 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 , , where is a path, and for . The cycle is said to have length ; moreover, two vertices and on are consecutive if either or . By we denote the set of vertices , and by we denote the partial hypergraph of .
A -cycle is a cycle of length at least 3 such that at most one vertex in belongs to three or more edges of .
A -cycle (a “weak -cycle” in [8]) is a cycle of length at least 3 such that every vertex in belongs to exactly two edges of .
A hypergraph is Berge-acyclic if it contains no cycles, -acyclic if it contains no -cycles, and -acyclic if it contains no -cycles (or, equivalently, if every partial hypergraph is -acyclic).
A reduced hypergraph is -acyclic if all its blocks are trivial hypergraphs. A hypergraph is -acyclic precisely if its reduction is -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, -acyclicity, -acyclicity, and -acyclicity exist, and the following is based on the 2-section of a hypergraph.
Proposition 2.4 (see [19]). A hypergraph is Berge-acyclic if and only if it is conformal and its 2-section is a block graph. A hypergraph is -acyclic if and only if it is conformal and its 2-section is a Ptolemaic graph. A hypergraph is -acyclic if and only if it is conformal and its 2-section is a strongly chordal graph. A hypergraph is -acyclic if and only if it is conformal and its 2-section is a chordal graph.
By Proposition 2.4, -acyclic hypergraphs are the same as “totally balanced” hypergraphs [3] and as “totally decomposable” hypergraphs in [12]. Finally, note that -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 -acyclic hypergraphs which in some sense represent the “superstructure” of a graph and of a hypergraph.
The prime hypergraph of graph is the (reduced) hypergraph whose edges are precisely the vertex sets of the prime components of .
Proposition 2.5 (see [21]). The prime hypergraph of a graph is a reduced, -acyclic hypergraph.
A nonempty subset of is a compact set of if is connected and no partial edge of is a separator. Note that if is a compact set, then has no articulation set; but the reverse does not hold in general (see Example 2.3). A compact component of is the reduction of the subhypergraph of induced by a maximal compact set. The compact hypergraph of is the (reduced) hypergraph whose edges are precisely the vertex sets of the compact components of .
Proposition 2.6 (see [22, 23]). The compact hypergraph of a hypergraph is a reduced, -acyclic hypergraph. Moreover, a hypergraph is -acyclic if and only if its reduction equals its compact hypergraph.
Example 2.3 (continued). The compact components of are shown in Figure 3 and the compact hypergraph of is .
3. 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 be a connected hypergraph. A set of subsets of is a convexity space [24, 25] if(i)the empty set, the singletons, and belong to ,(ii) is closed under set intersection,(iii)every set in is connected.
The sets in a convexity space are called the -convex sets of . For a subset of , the -convex hull of is the minimal (with respect to set inclusion) -convex set of that includes .
3.1. Graph Convexities
In this section, we recall the definitions and basic results on monophonic convexity (-convexity) and all-paths convexity (-convexity) on a connected graph [14, 17, 24–26]. Let be a connected graph. By and we denote the -convexity space and the -convexity space on , respectively.
A chord of a path is an edge of that joins two nonconsecutive vertices on . A path is chordless (or “induced” or “minimal”) if it has no chords. A subset of is -convex if, for every chordless path joining two vertices in , each vertex on belongs to .
The following result provides a known characterization of -convex sets. Let be a subset of ; an – path is a path with , .
Theorem 3.1 (see [5]). Let be a connected graph. A subset of is -convex if and only if, for every two distinct vertices and in , and are joined by an – path, then and are adjacent in .
Let be a connected graph with vertices and edges. Dourado et al. [25] gave an algorithm for computing the -convex hull of a subset of which runs in time. A better algorithm was given by Kannan and Changat [27], which runs in time.
A subset of is if, for every path joining two vertices in , each vertex on belongs to . It is easily seen that if is a tree, then the -convex hull of a subset of can be computed in time simply by deleting the leaves of that are not in .
We now state a characterization of those graphs on which = .
Theorem 3.2. Let be a connected graph.The equality holds if and only if is a tree.
Proof. Since every chordless path is trivially a path, the inclusion is obvious. If is a tree, then trivially one has . Assume that is not a tree. To prove that , consider any nontrivial block of , say . Let and be two adjacent vertices of . The path is the only chordless – path in and, hence, is -convex. On the other hand, is not -convex since, by Propositions 2.1 and 2.2, its -convex hull is , which proves that .
By Theorem 3.2, if is a tree, then the -convex hull of a subset of coincides with the -convex hull of and, hence, can be computed in time.
3.2. Hypergraph Convexities
Let be a connected hypergraph. In this section we recall the definitions and basic results on monophonic convexity (-convexity), canonical convexity (-convexity), simple-path convexity (-convexity), and all-paths convexity (-convexity) on . By , , , and we denote the -convexity space, the -convexity space, the -convexity space, and the -convexity space on , respectively.
3.2.1. -Convexity
Hypergraph -convexity [4] was not defined in terms of paths but using the hypergraph-theoretic version of Theorem 3.1, that is, a subset of is -convex if, for every two distinct vertices and in , and are joined by an – path, then and are adjacent in .
We now recall a useful characterization of -convex sets. To this end, we need further definitions.
Let be a subset of ; two edges and of are connected outside , written , if(i) or(ii) or(iii)there exists an edge of such that and .
The edge relation is an equivalence relation; the classes of the resultant partition of will be referred to as the -components of . The hypergraph is -connected if has exactly one -component. Note that is -connected if the graph is connected and no edge of is completely contained in . For an -component of , we call the set the boundary of with .
Example 3.3. Consider again the hypergraph of Example 2.3 (see Figure 1). For , the -components of are and the boundary of with is , and the boundary of with is , and the boundary of with is .
Theorem 3.4 (see [4]). Let be a connected hypergraph. A subset of is -convex if and only if either or the boundary of with every -component of is a clique of .
As we noted above, in contrast with -convexity in graphs, the original definition of -convexity in hypergraphs was given without having recourse to any path type. We now prove that -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 is a pair of nonconsecutive vertices on which are adjacent in . A path in is chordless if it has no chords.
Theorem 3.5. Let be a connected hypergraph. A subset of is -convex if and only if, for every chordless path joining two vertices in , each vertex on belongs to .
Proof. (if) Assume that contains for every chordless path joining two vertices in . By Theorem 3.4, it is sufficient to prove that the boundary of with every -component of is a clique of . Suppose, by contradiction, that there exists an -component of such that the boundary of with contains two vertices and that are not adjacent in . Let be the boundary of with . Since and are not adjacent in and are not adjacent in and, since is -connected, there exists a – path in of length at least 2 such that . Let be a – path of minimum length in . Of course, and is a chordless path; moreover, one has . Since and are not adjacent in must be of length at least 2 and, hence, there exists a vertex in that does not belong to . Since , does not belong to so that does not contain (contradiction).
(only if) Assume that is -convex. By Theorem 3.4, the boundary of with every -component of is a clique of . Suppose, by contradiction, that there exist two vertices and in and a chordless – path such that . Let and let . Of course, both and belong to the boundary of with the -component of containing and, since the boundary of with every -component of is a clique, and are adjacent. Since and are nonconsecutive on , the pair , is a chord of (contradiction).
Finally, let be a connected hypergraph with vertices and edges. An algorithm for computing -convex hulls in a hypergraph is the “monophonic-closure algorithm” [4] which runs in time if the prime hypergraph of is given. Since the time needed to construct the prime hypergraph of is [21], where is the number of edges of , and since , the time complexity of the monophonic-closure algorithm is . On the other hand, it is easy to check that is a chordless path in if and only if is a chordless path in , which implies that so that, given , the -convex hull of any vertex set in can be computed in time, that is, in time using the Kannan-Changat algorithm.
3.2.2. -Convexity
A subset of is -convex if the boundary of with every -component of is a partial edge of . Let be a connected hypergraph with vertices and edges. It is proven in [4] that -convex hulls can be computed using the Maier-Ullman algorithm [28], which runs in time. A more efficient algorithm is the “canonical-closure algorithm” [4], which runs in time if the compact hypergraph of is given. Since the time needed to construct the compact hypergraph of is [23], the time complexity of the canonical-closure algorithm is . Note that if is -acyclic, then the time complexity of the algorithm reduces to ; however, we can do better using the “selective-reduction algorithm” [18] which is linear in the size of .
3.2.3. -Convexity
A path in is simple [3] if every two nonconsecutive vertices on are not adjacent in the partial hypergraph ; equivalently, a path in is simple if is a chordless path in . A subset of is -convex if, for every simple path joining two vertices in , each vertex on belongs to . Let be a connected hypergraph with vertices and edges. An efficient algorithm for computing -convex hulls was given in [29], which runs in , where is the number of edges of the incidence graph of . Since , the time complexity of the algorithm is . However, if is -acyclic, then using the Anstee-Farber algorithm [15], the -convex hull of vertex set can be computed in time simply by deleting the “nest” vertices of that are not in (see [12]).
3.2.4. -Convexity
Let be a connected hypergraph. The convexity space is defined in the same way as in Section 3.1, that is, a subset of is -convex if, for every path joining two vertices in , each vertex on belongs to . Again one always has . In Section 6 we will give an efficient algorithm for computing -convex hulls.
4. -Convexity versus -Convexity
Since every partial edge is a clique, one always has . In this section we characterize the class of hypergraphs for which . First of all, observe that if is conformal, then every clique is a partial edge so that, by Theorem 3.4, every -convex set of is also -convex so that . We will see that conformality is not a necessary condition for .
A clique of is a boundary clique if there exists an -component of such that equals the boundary of with . A hypergraph is weakly conformal if every boundary-clique of is a partial edge of . Of course, every conformal hypergraph is weakly conformal. The following is an example of a weakly conformal hypergraph that is not conformal.
Example 4.1. Consider the (hyper)graph of Figure 4. The only cliques of that are not boundary cliques are the two cliques with cardinality 3, namely, the sets and . Since each clique of with cardinality less than 3 is a partial edge of , each boundary clique is a partial edge and, hence, is weakly conformal.
Theorem 4.2. Let be a connected hypergraph. The equality holds if and only if is weakly conformal.
Proof. (if) Let be any -convex set. Let be any -component of , and let be the boundary of with . By the very definition of -convexity, is a clique; moreover, is also a -component of and the boundary of with is itself. Therefore, is a boundary clique of . Since is weakly conformal, is a partial edge of . It follows that the boundary of with every -component of is a partial edge of and, hence, is -convex.
(only if) Let be any boundary clique of . Since is a clique, from the very definition of -convexity it follows that is -convex and, since by hypothesis, is -convex. From the very definition of -convexity, it follows that the boundary of with every -component of is a partial edge. Finally, since is a boundary clique of , there exists an -component of such that is the boundary of with . So, is a partial edge of . It follows that is weakly conformal.
5. -Convexity versus -Convexity and -Convexity
In this section we characterize the class of hypergraphs for which and the class of hypergraphs for which .
5.1. Equivalence between -Convexity and -Convexity
Let be a connected hypergraph. We first prove that . To achieve this, we need the following two technical lemmas.
Lemma 5.1. If is a – path in and and are not adjacent in , then there exists in a simple – path of length at least 2 and with .
Proof. Let be a – path. Let max and are adjacent in and let be an edge of that contains both and . Since and are not adjacent in , one has . Consider the – path . Let and are adjacent in , and let be any edge of that contains both and . If , then the – path is simple since and . If then let and are adjacent in , and let be any edge of that contains both are . If , then the – path is simple since and , and so on.
The next lemma characterizes -convex sets.
Lemma 5.2. A subset of is -convex if and only if either or, for every two distinct vertices and in , there exists no – path joining and in the partial hypergraph of obtained by deleting the edges that contain both and .
Proof. (only if) Assume that is -convex and . Suppose, by contradiction, that there exist two vertices and in and an – path joining and in . By construction of , the vertices and are not adjacent in so that by Lemma 5.1, there exists a simple – path of length at least 2 joining and in . Since is also a simple path in and is not contained in is not -convex (contradiction).
(if) If then is trivially -convex. Assume that and, for every two distinct vertices and in , there exists no – path joining and in . Suppose, by contradiction, that is not -convex. Then, there exist two vertices and in and a simple – path in with . Let be a vertex in , and let be the subpath of such that is on and is an – path. Of course,is a simple path in and has length at least 2. Let and be the vertices (in ) that are joined by . Since is a simple path in , no edge of contains both and , so that is also a path in . To sum up, is an – path that in joins the vertices and in so that is not -convex (contradiction).
Example 5.3. Let , where , , ,, , and . The hypergraph is shown in Figure 5. Let . Consider the three vertex pairs in . For the vertex pair , the partial hypergraph of is and, since 1 and 3 belong to distinct connected components of , there exists no – path joining 1 and 3 in . For the vertex pair , the partial hypergraph of is itself and, since every path joining 1 and 4 in passes through 3, there exists no – path joining 1 and 4 in . For the vertex pair , the partial hypergraph of is and, since 4 is not a vertex of , there exists no – path joining 3 and 4 in . By Lemma 5.2, the set is -convex, which is confirmed by the fact that the only simple paths joining two vertices in are , , , , and .
We first prove the inclusion .
Theorem 5.4. Let be a connected hypergraph. Every -convex set of is -convex.
Proof. Suppose, by contradiction, that there exists an -convex set of that is not -convex. Then, there exists an -component of such that the boundary of with (i.e., the set is not a partial edge of . Of course, and the boundary of with is not a partial edge of . Let be an edge of such that for every edge of , either or there exists . Since the boundary of with is not a partial edge of is a proper subset of ; therefore, there exist two vertices and such that and . Let be any edge of that contains . Since is an -component of , and, hence, there exists an – path in of length at least 2 (i.e., ) with . If no edge of contains both and , then is not -convex by Lemma 5.2 and a contradiction arises. Otherwise, let be the first edge on that contains both and . Since and , is not a subset of so that there exists . Consider the – path . If no edge of contains both and , then is not -convex by Lemma 5.2 and a contradiction arises. Otherwise, let be the first edge on that contains both and . Since and is not a subset of so that there exists . Consider the – path . If no edge of contains both and , then is not -convex by Lemma 5.2 and a contradiction arises, and so on. Thus, ultimately one obtains an – path of length at least 2 that joins two vertices and in and is such that no edge of contains both and . By Lemma 5.2, is not -convex and a contradiction arises.
We now characterize the class of hypergraphs for which .
Theorem 5.5. Let be a connected hypergraph. The equality holds if and only if is -acyclic.
Proof. (if) Assume that is -acyclic and suppose, by contradiction, that . Let be a -convex set that is not -convex. Then, there exist two vertices and in and a simple – path (of length at least 2) such that . Let , let be such that , let and , and let and . Consider the subpath of . Since is a simple path, is a simple path of length at least 2, and by construction. Let be the -component of such that . Then, both and belong to the boundary of with . Since is -convex, the boundary of with is contained in an edge of , say . Since is a simple path of length at least 2, and contains both and , is not an edge of so that is a cycle; moreover, since is a path of length at least 2, has length at least 3. Distinguish two cases depending on whether or not .Case 1 (). Then, ) is a cycle of length 3 and, since only the vertex belongs to the three edges of , is a -cycle (contradiction).Case 2 (). Then, there exists in a vertex for some , . Let and and and . Then is a cycle of length at least 3 and, since every vertex in belongs to exactly two edges of , is a -cycle (contradiction).
(only if) Assume that every -convex set of is also -convex and suppose, by contradiction, that is not -acyclic. Let , , be a -cycle. Distinguish two cases depending on whether or not each vertex in belongs to exactly two edges of .Case 1. Each vertex in belongs to exactly two edges of . Thus, is a -cycle. Let . Since , for each -component of is a partial edge of and, hence, is -convex. On the other hand, since each vertex in belongs to exactly two edges of , is simple path of length at least 2 and, since is not -convex (contradiction).Case 2. There exists a vertex in that belongs to more than two edges of . Without loss of generality, let it be . Since is a -cycle, each , , belongs to exactly two edges of . Let be any edge of , containing , and let , . Since , for each -component of is a partial edge of and, hence, is -convex. Let
It is easy to see that is a simple path of length at least 2 and, since is not -convex (contradiction).
Let be a connected hypergraph with vertices and edges. If is -acyclic, then is -acyclic and, hence, -convex hulls can be computed in using the Anstee-Farber algorithm. On the other hand, if is -acyclic, then is -acyclic and, hence, -convex hulls can be computed in linear time using the Tarjan-Yannakakis algorithm. By Theorem 5.5, -convex hulls can be computed in linear time, that is, more efficiently than using the Anstee-Farber algorithm.
5.2. Equivalence between sp-Convexity and -Convexity
Note that, by Theorem 3.5 and by the fact that every chordless path in is a simple path in , one always has . The following is another convexity-theoretic characterization of -acyclic hypergraphs.
Theorem 5.6. A connected hypergraph is γ-acyclic if and only if .
Proof. (only if) By hypothesis, is -acyclic so that by Theorem 5.5. Moreover, by Proposition 2.4, is conformal and, hence, weakly conformal, so that by Theorem 4.2. To sum up, .
(if) By hypothesis, . Since by Theorem 5.5 and , one has so that is -acyclic again by Theorem 5.5.
By Theorem 5.6, -convex hulls can be computed in linear time more efficiently than using the monophonic-closure algorithm.
6. -Convexity versus -Convexity, -Convexity, and -Convexity
Let be a connected hypergraph. In this section we characterize the three classes of hypergraphs for which , , and . To achieve this, we first give a polynomial algorithm for computing -convex hulls.
6.1. Computing -Convex Hulls
We represent by its incidence graph .
Remark 6.1. For every two vertices and of , every – path in is a – path in and vice versa; moreover, every cycle in is a cycle in and vice versa.
To avoid ambiguities, we call the vertices and edges of the nodes and arcs of , respectively. A node of is a vertex-node or an edge-node depending on whether or . Moreover, we call cutpoints the cut vertices of ; furthermore, a cutpoint of is a vertex-cutpoint or an edge-cutpoint depending on whether it is a vertex-node or an edge-node. Note that, if is a vertex-cutpoint of , then either the induced subhypergraph is not connected (see the vertex-node 3 in Figure 6) or the singleton is an edge of (see the vertex-node 1 in Figure 6); moreover, if is an edge-cutpoint of , then either the partial hypergraph is not connected (see the edge-node in Figure 6) or there exist one or more vertices of that belong to and to no other edge of (see the edge-node in Figure 6). Our algorithm works with the “block-cutpoint tree” of , which is defined as follows. Let be the bipartite graph whose nodes are the cutpoints and blocks of and where is an arc if the cutpoint of is a node of the block of . A node of is a block-node if it is a block of and a cutpoint-node otherwise. It is well known [14] that is a tree, which is called the block cut-vertex tree of . We also label each block-node of by the vertex set .
Example 6.1. Consider again the hypergraph of Example 5.3 (see Figure 5). The incidence graph of is shown in Figure 6, in which the cutpoints of are circled.
Note that the induced subhypergraph is connected and the induced subhypergraph is not connected; moreover, the partial hypergraph is not connected and the partial hypergraph is connected.
The blocks of are reported in Figure 7.
The block-cutpoint tree of is shown in Figure 8.
The six block-nodes of are labeled as follows:
Note that each leaf of is a block-node. Moreover, if is not a trivial hypergraph and is a one-point tree, then the node of is a nontrivial block of . Finally, if is a trivial block of with and , then the block-node of is not a leaf if and only if and are both cutpoints of .
Algorithm 1 constructs the -convex hull of any subset of .
| ||||||||||||||||
Example 6.1 (continued). When we apply Algorithm 1 with input , the tree resulting from the pruning of is shown in Figure 9. So, the output of Algorithm 1 is .
When we apply Algorithm 1 with input , the tree resulting from the pruning of is shown in Figure 10. So, the output of Algorithm 1 is .
Remark 6.2. Each time a block-node is deleted during the pruning process, either or and the vertex in belongs to for some undeleted block-node . Therefore, one has that is a subset of .
Fact 1. Each leaf of is a block-node and, if is a leaf of , then . and is a one-point tree, then the block-node of is a nontrivial block of .
Fact 2. Let be a block-node of that is a trivial block of with and . , then is not a one-point tree and, furthermore,(i)if then is a node of adjacent to (see the cutpoint-node and the block- node in Figure 7) for, otherwise, the block-node would be a leaf of and would be the only node of adjacent to and, since , the leaf of would have been deleted at step (1),(ii)if then is not a leaf of (see the block-node in Figure 10) for, otherwise, since , the block-node would have been deleted at step (1).
Fact 3. If a cutpoint-node of is a vertex-node of , say (see the cutpoint-node 3 in Figure 8), then there exist two leaves and of such that every –path in passes through the node . Furthermore, since (by Fact 1), there exists a vertex in for, otherwise, and, since is the node adjacent to the leaf , the leaf of would have been deleted at step 1. Analogously, there exists a vertex in . Finally, every –path in passes through .
Theorem 6.3. Let be a connected hypergraph and let be a subset of . Algorithm 1 correctly computes the -convex hull of .
Proof. Let be the output of Algorithm 1 and let denote the -convex hull of . If , then (see step (2)) and , which proves the statement. Assume that . We first prove that and, then, .Proof of . By Remark 6.2, it is sufficient to prove that is -convex. If then the statement is trivially true. Assume that and let be any vertex in . Since , belongs to for some block-node of that has been deleted during the pruning process. Let be the last of such block-nodes and let be the cutpoint-node adjacent to that is in every path joining and every node of the tree resulting from the pruning process. Note that if is a vertex-cutpoint of then, since is the block-node with that is last deleted, one has even if is cutpoint-node of (see Figure 11). Therefore, every path in joining to any vertex in must pass through and, hence, no path in joining two vertices in can pass through and, by Remark 6.1, the same holds in , which proves that is -convex.Proof of . Consider again the tree resulting from the pruning process. By Fact 1, each leaf of is a block-node. Let be the set of the leaves of . Thus, for each , and, if , then the vertex in is not the cutpoint-node adjacent to in . Let and let denote the -convex hull of . We now prove that . Consider the subgraph of formed by the block-nodes of . First of all, observe that any cutpoint of lies on some path joining two vertex-nodes in . It follows that, for every trivial block , the vertex in lies on some path joining two vertex-nodes in ; moreover, for every nontrivial block of , there exist at least two nodes of that lie on some path joining two vertex-nodes in and, by Proposition 2.1, each node of lies on some path joining two vertex-nodes in . Therefore, every vertex-node of lies on some path joining two vertex-nodes in , that is, . By step (2) of Algorithm 1, and, hence, . Finally, since , one has so that .
The following result, which is an immediate consequence of Theorem 6.3, will be used in the sequel.
Corollary 6.4. Let be a connected hypergraph. For every block of , the subsets of that are -convex are precisely the empty set, the singletons and .
6.2. Equivalence between -Convexity and -Convexity
Recall from Section 3.2.4 that one always has . We will prove that if and only if the incidence graph of satisfies the following two conditions:(C1) every edge-cutpoint of is a node of only trivial blocks of ;(C2) for every nontrivial block of , and for every with , there exist two distinct vertex-nodes and in and an – path joining and in the induced subgraph of obtained by deleting the edge-nodes that are adjacent to both and .
Note that every graph satisfies (C1) owing to the fact that every edge contains exactly two vertices.
Remark 6.5. If (C1) holds then, for every nontrivial block of , one has for, otherwise (i.e., if ), would contain an edge-cutpoint of (see Figure 12) owing to the fact that distinct edge-nodes of have distinct sets of adjacent vertex-nodes.
In order to characterize the class of hypergraphs for which , we first restate Lemma 5.2 as follows.
Lemma 6.6. Let be the incidence graph of . A subset of is -convex if and only if either or, for every two distinct vertex-nodes and in , there exists no – path joining and in the induced graph of obtained by deleting the edge-nodes adjacent to both and .
Corollary 6.7. Let be the incidence graph of . If condition (C2) holds then, for every block of , the subsets of that are -convex are precisely the empty set, the singletons and .
Proof. The empty set and the singletons are trivially -convex. Moreover, by Lemma 6.6 and condition (C2), for every nontrivial block of , no subset of with is -convex.
Theorem 6.8. Let be a connected graph. The equality holds if and only if the incidence graph of satisfies both conditions (C1) and (C2).
Proof. (only if) We first proveif then (C1)and, then,if then (C2).Proof of (i). Suppose, by contradiction, that condition (C1) does not hold. Then, there exists an edge-cutpoint of that is a node of a nontrivial block of . Let , let , and let be the block containing both and (see Figure 13).
By Theorem 6.3, the -convex hull of is , which is a proper superset of as is a nontrivial block of . On the other hand, since is the only simple – path in , the set is -convex, which contradicts the hypothesis .Proof of (ii). Let be any nontrivial block of . Note that, since (C1) holds, by Remark 6.5 one has . Let be any subset of with . By Corollary 6.4, the set is not -convex and, since , is not -convex. By Lemma 6.6, there exist two distinct vertex-nodes and in and an – path joining and in . Since , by Proposition 2.2 every path joining and in is also a path in ; therefore, there exists an – path joining and in , which proves that condition (C2) holds.
(if) Let be any subset of with , let and denote the -convex hull and the -convex hull of , respectively. Of course, . Therefore, it is sufficient to prove that . By Theorem 6.3, is the output of Algorithm 1, that is, if is the tree resulting from the pruning of the block-cutpoint tree of , then is the union of the sets for all block-nodes of . So, we need to prove that, for every block-node of , one has . Distinguish two cases depending on whether or not is a trivial block of .Case 1. is a trivial block of . Let be the unique vertex in . If then since . Consider now the case that . Since , Fact 1, the block-node of is not a leaf of . By Fact 2, is one of the nodes of adjacent to the block-node . By Fact 3, there exist two vertices and in such that every – path in passes through . Since every –path in is a –path in (see Remark 6.1), every simple –path in passes through so that belongs to and, hence, to , which proves that .Case 2. is a nontrivial block of . Consider first the case that is a one-point tree; thus, is the only node of and . Since and condition (C2) holds, by Corollary 6.7. Consider now the case that is not a one-point tree. Since is a nontrivial block, by (C1) the cutpoint-nodes of that are adjacent to the block-node are vertex-nodes of . Let us distinguish two cases depending on whether or not is a leaf of .Subcase 1. is a leaf of . Let be the cutpoint-node of that is adjacent to ; thus, . By Fact 3, there exist a vertex in and a vertex in , where is another leaf of , such that every – path in passes through . Since every –path in is a –path in (see Remark 6.1), every simple – path in passes through so that belongs to and, hence, to . Since both and belong to , one has . If one had then, since condition (C2) holds, by Corollary 6.7 the set would not be -convex and, hence, there would exist a vertex in that lies on some simple path joining two vertices in which contradicts the fact that the set is -convex. Therefore, one has .Subcase 2. is not a leaf of . Let and be two cutpoint-nodes of that are adjacent to ; thus, . By Fact 3, for each there exist two distinct vertices and in such that every – path in passes through . Since every – path in is a – path in (see Remark 6.1), every simple – path in passes through so that belongs to and, hence, to . Since both and belong to both and , one has . If one had , then, again by Corollary 6.7, the set would not be -convex and, hence, there would exist a vertex in that lies on some simple path joining two vertices in which contradicts the fact that is an -convex set. Therefore, one has .
6.3. Equivalence between -Convexity and -Convexity
The next result characterizes the class of hypergraphs for which .
Theorem 6.9. Let be a connected hypergraph. The equality holds if and only if is Berge-acyclic.
Proof. By Remark 6.1, is Berge-acyclic if and only if is a tree.
(only if) Assume that and suppose, by contradiction, that is not Berge-acyclic. Since is not a tree, there exists a nontrivial block of , say . Then there exists an edge-node of such that (see Figure 14). By Corollary 6.4, is not -convex; but, since is an edge of , is -convex (contradiction).
(if) Assume that is Berge-acyclic. Then, is a tree and, hence, every block of is trivial so that trivially conditions (C1) and (C2) of Theorem 6.8 hold and, hence, . Moreover, since every Berge-acyclic hypergraph is -acyclic, is -acyclic so that by Theorem 5.5. To sum up, one has , which proves the statement.
If is Berge-acyclic, then is -acyclic and, hence, -convex hulls can be computed in linear time. By Theorem 6.9, -convex hulls can be computed in in linear time more efficiently than using Algorithm 1.
6.4. Equivalence between -Convexity and -Convexity
The next theorem generalizes Theorem 3.2.
Theorem 6.10. Let be a connected hypergraph. The equality holds if and only if is Berge-acyclic.
Proof. If is Berge-acyclic, then, by Proposition 2.4, is conformal and, by Theorem 4.2, . By Theorem 6.9, = and, hence, . On the other hand, if then, since , one has so that, by Theorem 6.9, is Berge-acyclic.
7. Future Research
We have considered the four notions of hypergraph convexity: -convexity, -convexity, -convexity, and -convexity, and for each pair of these convexities, we have characterized the class of hypergraphs in which the two convexities are equivalent. Another important notion of graph convexity is geodetic convexity (or -convexity) [3]: a vertex set in a graph is -convex if, for every geodesic (i.e., minimum-length path) joining two vertices in , each vertex on belongs to . Of course, since every geodesic is a chordless path, every -convex set is -convex, that is, -convexity is finer than -convexity; moreover, -convexity and -convexity are equivalent in distance-hereditary graphs [3]. An open problem is the characterization of graphs (and, more in general, of hypergraphs) in which -convexity and -convexity are equivalent.