Clustered Spanning Tree - Conditions for Feasibility

Let H = < V, S > be a hypergraph, V is a set of vertices and S = { S 1 , . . . , S m } is a set of not necessarily disjoint clusters S i ⊆ V such that ∪ mi =1 S i = V . The Clustered Spanning Tree problem is to ﬁnd a tree spanning all the vertices of V which satisﬁes that each cluster induces a subtree, when it exists. We provide an efﬁcient and unique algorithm which ﬁnds a feasible solution tree for H when it exists, or states that no feasible solution exists. The paper also uses special structures of the intersection graph of H to construct a feasible solution more efﬁciently. For cases when the hypergraph does not have a feasible solution tree, we consider adding vertices to exactly one cluster in order to gain feasibility. We characterize when such addition can gain feasibility, ﬁnd the appropriate cluster and a possible set of vertices to be added.


Introduction
Let H =< V, S > be a hypergraph, where V is a set of vertices and S is a set of clusters S 1 , . . ., S m , S i ⊆ V for i ∈ {1, . . ., m}, ∪ m i=1 S i = V , and the clusters are not necessarily disjoint.The Clustered Spanning Tree problem, denoted by CST, is to find a tree spanning all the vertices of V , such that each cluster induces a subtree, if one exists.
One of the main results of this paper (Algorithm ES, Figure 1) is a novel algorithm for the essential question of whether a feasible solution tree exists for a given instance of the CST problem.This algorithm requires O(|V | 2 m) time complexity and can handle every instance hypergraph.In the first stage of the algorithm a weighted graph is constructed, V is the set of vertices of this graph, an edge (v, u) exists in the graph if there is a cluster which contains both vertices, the weight of each edge in the graph is equal to the number of clusters containing both endpoints of this edge.Next, we find a maximum spanning tree for this graph.A feasible solution for the CST problem exists if and only if the weight of this tree is When the intersection graph contains a cut-edge, we prove that deciding whether a feasible solution exists can be based on the decision made independently for each component of the intersection graph.The feasible solution tree for the given hypergraph is constructed using the feasible solution subtrees created from the corresponding subproblems and thus this technique may significantly reduce the required complexity.For the special case where every vertex in V is contained in at most 2 clusters from S, the CST problem has a feasible solution if and only if the corresponding intersection graph is a tree.In all cases, a feasible solution tree is offered when it exists.
Another main result of the paper (Theorems 4.12 and 4.15) considers hypergraphs which do not have a feasible solution tree.In these cases we want to characterize when adding vertices to exactly one cluster can gain feasibility.Assuming the clusters satisfy the Helly Property, we prove that when all the chordless cycles of the intersection graph share a joint node, adding an appropriate set of vertices to the corresponding cluster creates a hypergraph with a feasible solution tree.We classify all the sets of vertices whose addition creates feasibility.
Throughout this paper, we assume that the intersection graph of H is connected.Otherwise, a feasible solution tree for H can be constructed by properly adding edges between the feasible solutions of each connected component, if they exist.
The following theorem, summarized in McKee and McMorris (1999), which will be used throughout the paper, gives sufficient and necessary conditions for the feasibility decision problem for a given instance of the CST problem.
Theorem 1.1 (Duchet (1976), Flament (1978), Slater (1978)) A hypergraph H =< V, S > has a feasible solution tree if and only if it satisfies the Helly property and its intersection graph is chordal.
Verifying whether a hypergraph satisfies the Helly Property requires O(|V | 4 m) time complexity, according to Dourado et al. (2009).This time complexity dominates the time required to verify that the intersection graph is chordal.Thus using Theorem 1.1 to check whether a hypergraph has a feasible solution tree requires O(|V | 4 m) time complexity.
Related problems consider different structures of the solution tree and the clusters' induced subtrees.In Swaminathan and Wagner (1994) a polynomial algorithm is presented, which constructs a tree where each cluster spans a path.The most restricted problem where both the tree and subtrees are required to be paths, is in fact the Consecutive Ones Problem, which Booth and Lueker (1976) solve in linear time using PQ-trees.
When considering the optimization CST problem, the edges of E have weights and the objective is to find a feasible solution tree with minimum weight.This problem was solved in Korach and Stern (2003) where an optimum solution is found in O(|V | 4 m 2 ) time complexity, when a feasible solution exists.In addition, an abstraction of the problem using matroids is presented.For the restricted case where each cluster contains at most three vertices, there is a linear time algorithm and a polyhedral description of all feasible solutions.A special case of the optimization CST problem, where the optimum spanning tree solution is required to span a complete star on each cluster, is presented in Korach and Stern (2008).A structure theorem which describes all feasible solutions and a polynomial algorithm for finding an optimum solution is presented, assuming the intersection graph is connected.
Another related optimization problem, called the clustering-TSP-path, arises when the optimum solution tree is required to be a TSP-path.The TSP-path is proven to be NP-hard in Christofides (1976).In Guttmann-Beck et al. (2018) several algorithms for the not necessarily disjoint clustered TSP-path are presented.For a restricted case of the problem an exact polynomial algorithm is presented.For other cases approximation algorithms are presented, whose efficienty depends on the structure of the hypergraph.A lot of research has also been investigated on the clustered TSP-path where the clusters are disjoint.A heuristic for this problem is presented in Chisman (1975), a branch and bound algorithm for solving this problem is presented in Lokin (1979) and bounded-approximation algorithms are presented in Arkin et al. (1997) andGuttmann-Beck et al. (2000).In Anily et al. (1999) the ordered disjoint clustered TSP is considered and an approximation algorithm is offered.In Potvin and Guertin (1998) a genetic algorithm for solving this problem is presented.
This paper is organized as follows.Section 2 introduces Algorithm Existence of Solution which decides whether a feasible solution tree exists for any given instance of the CST problem and finds a feasible solution when one exists.Section 3 contains theorems which state how the feasibility of subproblems of a given instance indicates whether a feasible solution tree exists.This section also uses information derived from the intersection graph of a given instance to determine its feasibility.When applicable, the methods introduced in this section have significantly better complexity than the algorithm introduced in Section 2. Section 4 considers hypergraphs with no feasible solution tree.For those cases, we characterize when adding vertices to exactly one cluster of the given hypergraph gains feasibility.The section contains proofs which characterize the hypergraphs that become feasible when such an addition is preformed, identify which clusters are appropriate for this change and what are the relevant sets of vertices whose addition creates feasibility.

Existence of Solution
This section introduces algorithm Existence of Solution (ES) which is a major result of the paper.The algorithm, presented in Figure 1, either finds a feasible solution tree for a hypergraph H =< V, S > or declares that there is no feasible solution.The algorithm creates a weighted graph denoted by G ES = (V ES , E ES ), where V ES = V and E ES contains an edge (v, u) if there exists a cluster S i such that {v, u} ⊆ S i .The weight of each edge is equal to the number of clusters containing both endpoints of the edge.In this graph a maximum spanning tree T ES is found.We prove that a hypergraph has a feasible solution tree if and only if the weight of T ES , denoted by w(T ES ), is equal to m i=1 |S i | − m.When a solution exists, T ES is a feasible solution tree.
First, we define induced subtrees and the weighted graph G ES which is used by Algorithm ES.
Definition 2.1 Given a hypergraph H =< V, S >, T a tree spanning V and V ⊆ V , the subgraph of T induced by V , denoted by T [V ], is defined to contain all the vertices of V and all the edges of T whose both endpoints are in V .
Definition 2.2 Given a hypergraph H =< V, S >: • G ES is the weighted graph with vertex set V ES and edge set E ES , where: • For every edge (v, u) ∈ E ES and every cluster S i ∈ S: . ., m}, {v, u} ⊆ S i }|.This is the weight used in Algorithm ES (Figure 1).
Lemma 2.3 Given a hypergraph H =< V, S >, the inequality w i (T ) ≤ |S i |−1 holds for every spanning tree T of G ES and every cluster S i ∈ S.An equality holds if and only if T [S i ] is a subtree spanning all the vertices in S i .
Proof: According to Definition 2.2, the weight w i (T ) is equal to the number of edges in is a spanning tree of S i for every cluster S i ∈ S. Therefore, T ES is a feasible solution tree for H. 2 In Figure 2 we give two examples of hypergraphs.The left most item in this figure is the hypergraph suited for S = {{1, 2}, {2, 4}, {1, 2, 3}, {2, 3, 4}}.The second item on the left is the appropriate weighted graph created for this hypergraph by Algorithm ES.The solid lines describe the maximum spanning tree of this graph.The weight of this tree is 6 =

Induced Hypergraphs
Consider a hypergraph H =< V, S > which is an instance for the CST problem.New instances of the problem are created by considering induced hypergraphs defined by subsets of S. In this section we prove that when H has a feasible solution tree T H , then induced subtrees of T H are feasible solution trees for the corresponding induced subproblems.Furthermore, when an induced hypergraph does not have a feasible solution tree, neither does H. Section 3.1 considers the case where the intersection graph of H has a cut-edge.In this case, we prove that the hypergraph has a feasible solution tree if and only if the induced hypergraphs, corresponding to the connected components created by removing the cut-edge, have feasible solution trees.Furthermore, a feasible solution tree for H can be constructed from the solution trees of the induced hypergraphs.Section 3.2 considers the special case where each vertex of V is contained in at most two clusters.In this case, we prove that H has a feasible solution tree if and only if the corresponding intersection graph is a tree.
The following definitions will be used throughout the paper.
Definition 3.1 Given a hypergraph H =< V, S > where S is a set of not necessarily disjoint clusters {S i1 , . . ., S ip } of V .The intersection graph of {S i1 , . . ., S ip }, denoted by G int ({S i1 , . . ., S ip }), is defined to be a graph whose set of nodes is {s i1 , . . ., s ip }, where s ij corresponds to S ij , and an edge Definition 3.2 Let H =< V, S > be a hypergraph and let S ⊂ S be a set of clusters.We define H[S ] to be the hypergraph whose vertex set is V (S ) = Si∈S S i and its clusters set is S .Proof: First we prove that T H [V (S )] is a tree.Since T H [V (S )] is a subgraph of T H clearly it has no cycles.
To prove that this subgraph is connected, consider v 1 , v 2 ∈ V (S ).There are two clusters S i1 , S i2 ∈ S , not necessarily distinct, such that v 1 ∈ S i1 and v 2 ∈ S i2 .
If S i1 = S i2 then since G int (S ) is connected it contains a simple path (s i1 , w 1 ), (w 1 , w 2 ), . . ., (w k , s i2 ) with s i1 representing S i1 , s i2 representing S i2 and w i representing a cluster W i ∈ S , ∀i ∈ {1, . . ., k}.Hence, there are vertices u 1 , . . ., u k+1 ∈ V (S ) such that: Since T H is a feasible solution tree it contains the following simple paths: 1. a path v 1 − u 1 whose vertices are all contained in S i1 .
2. ∀i ∈ {2, . . ., k} a path u i−1 − u i whose vertices are all contained in W i , 3. a path u k+1 − v 2 whose vertices are all contained in S i2 .
T H [V (S )] contains all these paths and their union gives a path connecting Similarly, if S i1 = S i2 then T H contains a path connecting v 1 and v 2 whose vertices are all contained in S i1 which is also in Combining the above claims we get that T H [V (S )] is cycle-less and connected and therefore a tree.Since T H is a feasible solution tree, the induced tree T H [V (S )] on any S i ∈ S is equal to the induced tree of T H on S i and hence is a subtree spanning S i .Thus, T H [V (S )] is a feasible solution tree for H[S ].
Corollary 3.5 Let H =< V, S > be a hypergraph with a connected intersection graph G int (S).If G int (S ) is connected for S ⊂ S and H[S ] has no feasible solution tree, then H has no feasible solution tree.

Breaking the Intersection Graph
This section introduces a strategy to decide whether it is possible to divide the problem into smaller subproblems by a cut-edge of the intersection graph.We prove that a feasible solution tree of the given problem exists if and only if each subproblem has a feasible solution tree.Since the instances for the subproblems may be significantly smaller, the complexity of deciding whether a subproblem has a feasible solution decreases according to the size of the subproblem.
Theorem 3.6 Let H =< V, S > be a hypergraph with a connected intersection graph G int (S).If the intersection graph contains a cut-edge (bridge) which divides the intersection graph into two connected components G int (S ) and G int (S\S ), then H has a feasible solution tree if and only if each one of H[S ] and H[(S\S )] has a feasible solution tree.
Proof: Without loss of generality, denote the cut-edge as (s 1 , s 2 ) with s 1 representing cluster S 1 ∈ S and s 2 representing cluster S 2 ∈ S\S .According to the theorem's assumption If H has a feasible solution tree then since both G int (S ) and G int (S\S ) are connected, according to Theorem 3.4, H[S ] and H[S\S ] both have feasible solution trees.
Suppose, on the other end, that H[S ] and H[S\S ] have feasible solution trees T 1 and T 2 respectively.Since T 1 is a feasible solution tree for H[S ], T 1 [S 1 ] is a connected subtree.According to the theorem's assumption, the only edge in G int between S and S\S is (s 1 , s 2 ).Therefore, (S 1 ∩S 2 )∩V (S\{S 1 , S 2 }) = φ.Hence, the vertices contained in T 1 [S 1 ] can be reordered to obtain a connected subtree on T 1 [S 1 ∩ S 2 ].T 2 can be reordered in a similar way to satisfy that T 2 [S 1 ∩ S 2 ] is a connected subtree of T 2 [S 2 ].This can be performed also to satisfy T 2 [S 1 ∩ S 2 ] = T 1 [S 1 ∩ S 2 ].After the described adjustments, the union of T 1 and T 2 is a feasible solution tree for H. 2

Bounded Containment of Vertices
In this section a special type of hypergraphs is considered, when every vertex in V is contained in at most 2 clusters from S. We prove that in this type of hypergraphs, the CST problem has a feasible solution if and only if the intersection graph is a tree.First, some definitions and properties are presented.
Observation 3.8 In an intersection graph, a vertex v ∈ V which belongs to k clusters (nc(v) = k) creates a k-size clique.Hence, if the intersection graph of a hypergraph is a tree or a chordless cycle with at least 4 nodes, then nc(v) ≤ 2 ∀v ∈ V .
• For every i = j ∈ {1, . . ., m}, define the subcluster K {i,j} = {v : v ∈ S i ∩ S j }.K {i,j} contains all the vertices which belong both to S i and to S j .
The following Property follows from Definition 3.9.
Theorem 3.11 Let H =< V, S > be a hypergraph with nc(v) ≤ 2 ∀v ∈ V and a connected intersection graph.H has a feasible solution tree if and only if its intersection graph is a tree.
A maximum spanning tree T ES contains a maximum number of edges of weight 2. Inside each subcluster K {i,j} a subtree is chosen, using |K In an intersection graph, the degree of s i (the vertex representing S i ) is By Property 3.10 each vertex of the graph belongs to exactly one subcluster, hence: Using Equations ( 2) and ( 3) and Property 3.10: According to Theorem 2.5, hypergraph H has a feasible solution tree if and only if 4), hypergraph H has a feasible solution tree if and only if Since d i is also the degree of s i in the intersection graph, 0.5 m i=1 d i is equal to the number of edges of this intersection graph.The number of edges in a connected intersection graph with m nodes is equal to m − 1 if and only if the intersection graph is a tree, thus proving the correctness of the theorem. 2 Theorem 3.11 proves the following remark, which can also be deduced from Theorem 1.1: 4 Inserting Vertices to Gain Feasibility In this section we discuss hypergraphs with no feasible solution tree.We consider adding vertices to only one cluster of the given hypergraph.We characterize the hypergraphs that become feasible when such an addition is preformed.We find an appropriate cluster and all the set of vertices which could be added to it.
Definition 4.1 Let H =< V, S > with S = {S 1 , . . ., S m }, be a hypergraph with no feasible solution tree.An insertion cluster is a cluster S i ∈ S, such that there exists a set of vertices U ⊂ V , for which the hypergraph for every set of clusters {S i1 , . . ., S iq } ⊆ S , when q j=1 S ij = φ, for q ≥ 2, there is at least one node u ∈ U such that u ∈ q j=1 S ij .In this case we say that u covers the intersection q j=1 S ij .Definition 4.3 Let H =< V, S > be a hypergraph, H is a cycled hypergraph if the clusters in S satisfy the Helly Property and if G int (S) is composed only of chordless cycles, each one of size ≥ 4.
of S\{S i }, there exists u ∈ U such that u ∈ S j ∩ S k .According to the way S U j and S U k are defined, According to the way these clusters were defined S j ⊇ S U j and Lemma 4.7 In Algorithm IC (Figure 3),  3) is a tree which spans V .
Proof: In the algorithm, T is initialized to be a feasible solution tree of H U .In this stage T is a spanning tree of the vertices in U .Consider a vertex v ∈ V \U .There are two options: 1. v ∈ ( j =i S j )\U , in this case the algorithm finds the set of indices M I(v, i) and a vertex u ∈ U such that u ∈ j∈M I(v,i) S j .Vertex u exists since U is an intersection cover set.Next, the algorithm adds v to T as a leaf.
2. v ∈ S i \(U ∪ ( j =i S j )), in this case v is added to T as a leaf in the last step of the algorithm.
Hence, T is a tree which touches all the vertices of V . 2 Lemma 4.9 In the output tree T returned by Algorithm IC (Figure 3) , T [S k ] is a connected subtree for every k ∈ {1, . . ., m}, k = i.
Proof: In Algorithm IC, T is initialized to be a feasible solution tree of H U .Therefore, at this stage T [S U k ] is a subtree spanning S k ∩ U .In all the following steps of the algorithm T [S U k ] does not change.Consider a vertex v ∈ S k \U , in this case v ∈ (( j =i S j )\U ).
If |M I(v, i)| = 1 then S k is the only cluster to contain v. Since G int (S) is connected, S k intersects with at least one other cluster S j = S i .U is an intersection cover set, therefore there exists u ∈ U which covers S j ∩ S k , proving that Algorithm IC can find u ∈ S U k .If |M I(v, i)| > 1 and since U is an intersection cover set of S\{S i }, there exists u ∈ U which covers j∈M I(v,i) S j .Since j∈M I(v,i) S j ⊂ S k , it follows that also in this case u ∈ S U k .In both cases Algorithm IC connects v to T [S U k ] by an edge (v, u) with u ∈ S U k .At the end of the algorithm, all the vertices of S k are either in T [S U k ] or connected to it by an edge whose both endpoints are in S k , thus proving the required subtree. 2 Proof: First we prove that S j ∩ U = φ ∀j ∈ {1, . . ., m}, j = i.Suppose by contradiction that there exists some cluster S j with S j ∩ U = φ.Consider a cycle C in G int (S) containing S j .Since H is a cycled hypergraph, its clusters satisfy the Helly Property and C contains at least 4 nodes.Let S j l and S jr be two clusters in C (not necessarily distinct), each one on a different path in C between S i and S j .Choose S j l (respectively Sj r ) to be the cluster closest to S j which satisfy S j l ∩ U = φ ( S jr ∩ U = φ) if it exists, otherwise let S j l = S i (S jr = S i ).Let s i be the node representing S i ∪ U in the intersection graph of H .This intersection graph contains the cycle According to the Lemma's assumption, H has a feasible solution tree.Therefore, according to Theorem 1.1, C contains exactly 3 nodes.Without loss of generality, suppose that C contains nodes s i = s j l , s j and s jr .According to Theorem 1.1, the clusters of H satisfy the Helly Property, therefore there exists v ∈ (S i ∪ U ) ∩ S j ∩ S jr .Since S j ∩ U = φ it follows that v ∈ S i ∩ S j ∩ S jr , contradicting the fact that cycle C contains at least 4 nodes.Now we prove that U contains a vertex which covers each intersection.Consider q ≥ 2 distinct clusters S j1 , . . ., S jq for {j 1 , . . ., j q } ⊆ {1, . . ., i − 1, i + 1, . . ., m} with S j1 ∩ . . .∩ S jq = φ and S j1 ∩ . . .∩ S jq ⊂ S i .Since S j1 ∩ U = φ, . . ., S jq ∩ U = φ, in H the q + 1 clusters S j1 , . . ., S jq , S i ∪ U pairwise intersect.Since the clusters in H satisfy the Helly Property S j1 ∩. ..∩S jq ∩(S i ∪U ) = φ giving that S j1 ∩ . . .∩ S jq ∩ U = φ.Thus, there exists u ∈ U which covers the intersection S j1 ∩ . . .∩ S jq . 2 Theorem 4.15 Let H =< V, S > be a hypergraph with no feasible solution tree.Suppose the clusters of H satisfy the Helly Property and there is at least one node s i which is contained in every chordless cycle of G int (S).If adding U ⊂ V to S i creates a hypergraph which has a feasible solution tree, then U is an intersection cover set of all the clusters, excluding S i , which are contained in the chordless cycles in the intersection graph.
Proof: Similar to the proof of Theorem 4.12, using Lemma 4.14. 2 According to Theorems 4.12 and 4.15, when H =< V, S > is a hypergraph whose clusters satisfy the Helly Property and s i is a node which is contained in all the chordless cycles of G int (S), then S i is an insertion cluster.Adding the vertex set U ⊂ V to S i creates a hypergraph with a feasible solution tree if and only if U is an intersection cover set of all the clusters which are contained in the chordless cycles, excluding S i .Thus, finding the minimum number of added vertices to S i is equivalent to finding the minimum cardinality intersection cover set.Since the hypergraph satisfy the Helly Property, this is equivalent to finding all the maximal cliques of the intersection graph induced on the chordless cycles, which may require exponential complexity in the general case.However, for the more common case when nc(v) ≤ k, for a constant k, for all vertices in the hypergraph induced on the chordless cycles, the question of finding minimum cardinality intersection cover set becomes polynomial.For example, if G int (S) contains exactly one chordless cycle of p nodes, every vertex v of the induced hypergraph on this cycle satisfies that nc(v) ≤ 2. In this case, the minimum intersection cover set will contain p − 2 vertices.
All the above discussion assumes that the clusters satisfy the Helly Property.Adding vertices to an insertion cluster may increase nc(v) of a vertex v by at most 1.Therefore, if the intersection graph contains a clique on the nodes s i1 , s i2 , . . ., s ip and max{nc(v)|v ∈ ∪ q j=1 S ij } < p−1 then the hypergraph has no insertion cluster.Furthermore, if the clusters do not satisfy the Helly Property, H has S i as an insertion cluster if the following two conditions are satisfied: • In the intersection graph the corresponding node s i is contained in all the chordless cycles.
• Adding vertices to S i causes the clusters of every clique of the intersection graph to satisfy the Helly Property.

Summary and Further Research
In this paper we introduce different algorithms for the CST problem.The first algorithm introduced in this paper creates a weighted graph where the weight of the maximum spanning tree of this graph indicates whether a feasible solution tree exists.Furthermore, the maximum spanning tree offers a feasible solution, if one exists.The other methods introduced in this paper decide whether a feasible solution exists and find such a tree when it exists, for some special structures of the intersection graph.
For those instances where no feasible solution tree exists, we characterize when adding vertices to exactly one cluster will gain feasibility.This approach finds the appropriate cluster and the vertices that should be added.Further research may be applied to find possible vertices insertion to more than one cluster of the given hypergraph.It is of special interest to define which insertions to perform and how to measure their minimality.
returnsA feasible solution tree or a statement "No feasible solution".begin Construct the following weighted graph m i=1 |S i | − m which indicates, according to Theorem 2.5, that the hypergraph has a feasible solution tree with the maximum spanning tree as its solution.The two right items of this figure describe the hypergraph and the weighted graph suited for S = {{1, 2}, {2, 4}, {1, 2, 3}, {1, 3, 4}}.The weight of a maximum spanning in this case is 5 = m i=1 |S i | − m and indeed the hypergraph has no feasible solution tree.

Fig. 2 :
Fig. 2: Example of a hypergraph (a) with a feasible solution tree and (b) without a feasible solution tree {i,j} | − 1 edges.By property 3.10, these subclusters are disjoint, giving that altogether T ES contains m i=1 m j>i K {i,j} =φ (|K {i,j} | − 1) edges of weight 2. The rest of the edges are of weight 1.Since any spanning tree on n vertices contains n − 1 edges, there are (n − 1) − ( =φ (|K {i,j} | − 1)) edges of weight 1.Hence, the weight of a maximum spanning tree returned by Algorithm ES: w(T ES ) = 2 * ( =φ (|K {i,j} | − 1)) + 1 * ((n − 1) − ( contains a chordless cycle with at least 4 nodes, then H has no feasible solution tree.Proof: Denote by s i1 , . . ., s ip , p ≥ 4, the nodes of a chordless cycle in G int (S) and let S C = {S i1 , . . ., S ip }.Consider the induced hypergraph H[S C ], its intersection graph G int (S C ) is a chordless cycle and therefore it is connected.According to Observation 3.8, ∀v ∈ V (S C ) it holds that nc(v) ≤ 2. By Theorem 3.11, since G int (S C ) is a cycle, H[S C ] has no feasible solution tree.Therefore, by Corollary 3.5, H has no feasible solution tree.2 is the intersection graph of H, then the induced graph G int (S)[( Si∈S S i )] is the intersection graph of H[S ] and therefore can be denoted as G int (S ).
Theorem 3.4 Let H =< V, S > be a hypergraph with a connected intersection graph G int (S) and a feasible solution tree T H .If G int (S ) is connected for S ⊂ S, then T H [V (S )] is a feasible solution tree for H[S ].