Clique trees of infinite locally finite chordal graphs

We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting.


Introduction
A chordal graph is a graph, where every cycle of length greater than three contains a chord, i.e. an edge connecting two non-consecutive vertices along the cycle. Chordal graphs are a classic object in graph theory and computer science [BP93]. They are equivalent to the class of graphs representable as a family of subtrees of a tree [Gav74,Hal84]. Each finite and connected chordal graph has natural representations of this form, with the trees being a subclass of the spanning graphs of its clique graph. These trees are called clique trees. There are a number of characterisations of clique trees among all spanning trees of the clique graph. They relate various properties of a clique tree to minimal vertex separators of the original graph, or maximality with respect to particular edge weights in the clique graph, or properties of paths in the tree, among others.
The present paper investigates clique trees of infinite, locally finite chordal graphs. We first prove the existence of at least one clique tree. Classic proofs of the various properties of clique trees often rely heavily on the finiteness of the setting. All of the known characterising properties are either not sensible in the infinite setting (as the maximality with respect to edge weights), or are of unbounded range (running intersection property of paths), or have at least overlapping constraints.
Our core contribution is a local partition of the edge set of the clique graph and a corresponding set of constraints, one for each element of the partition, which a clique tree has to fulfil. Each constraint only depends on the edges within its partition element, whence the constraints can be satisfied or violated independently from each other. This allows a local construction of a clique tree by fixing a satisfying subset of the edges in each element of the partition. A characterisation of all clique trees is possible via a bijection with the product of the local choices.
In the case of a finite chordal graph, our characterisation permits an enumeration of the clique trees. It turns out that this enumeration is equivalent to a prior enumeration via a local partitioning of constraints by Ho and Lee [HL89]. Their partition is indexed by the minimal vertex separators of the chordal graph. We use a different approach based on families of cliques and recover the minimal vertex separators a posteriori. Specifically, the intersections of the cliques in a clique family is a minimal vertex separator, and vice-versa.
We derive classic properties of clique trees for infinite graphs from the above local decomposition property. A finer analysis of the structures appearing in the local decomposition points out connections with minimal vertex separators and the reduced clique graph [GHP95].
The structure of this paper is as follows: Section 2 introduces basic notation, clique trees and clique families. Section 3 contains our existence and characterisation theorems for clique trees. Section 4 discusses counting and enumerating the clique trees and section 5 derives the classic properties of clique trees. Section 6 contains the proofs of the statements from section 3.

Notation and basic properties 2.1 Graphs
Throughout the present work, we only consider locally finite graphs. Let G = (V, E) be a graph and W ⊆ V . Denote by G[W ] the induced subgraph of G with vertex set W . Contracting the set W into a single vertex yields the graph G/W . It may contain multiple edges and loops, even if the graph G did not. If V 1 , V 2 , . . . , V k are disjoint subsets of V , then G/{V 1 , V 2 , . . . , V k } denotes the graph resulting from G by contracting each V i to a single vertex, where the order of contractions has no influence on the final result. For an equivalence relation ∼ on V , denote by G/ ∼ the graph resulting from contracting each equivalence class with respect to ∼.
We call a set finite W ⊆ V complete, iff G[W ] is a complete graph on W . A clique is a maximal complete set of vertices of G. Denote by C G the set of complete subsets of V and by M G the set of all cliques of G. The clique graph M G of G has vertex set M G and an edge for every pair of cliques with non-empty intersection.
A tree T is a connected and acyclic graph. For two vertices v, w ∈ T , there is a unique path P T (v, w) in T . A subgraph of G is spanning, iff it has the same vertex set as G. The set of spanning trees of G is T G . We admit the empty graph, which is a graph without vertices, and consider it a tree. Also, the only spanning tree of the empty graph is the empty graph.

Chordal graphs and subtree representations
Our main reference for basic facts about chordal graphs is [BP93]. A chordal graph has no cycle of length greater than 3. In other words, every closed path of length greater than 3 has a chord, an edge connecting two non-consecutive vertices of the cycle. Throughout this work, we assume that chordal graphs are connected.
Let T be a tree and denote by T the family of subtrees of T . A function A graph is chordal, iff is has a subtree representation on some tree [Gav74,Hal84]. This does not hold for general countable, non locally-finite graphs [Hal84]. If G is finite, there is combinatorial representation [Gav74], where T is a spanning tree of M G and t(v) We call T a clique tree of the chordal graph G. The set of all clique trees TC G of G is the set of all all spanning trees T ∈ T M G fulfilling (1)

The lattice of clique families
Let W ⊆ V . The clique family generated by W is The set of clique families associated with G is Generation is anti-monotone: If W / ∈ C G , then K(W ) = ∅. The largest clique family is K(∅) = M G . It is infinite and the only infinite clique family, iff G is infinite itself. The set of finite clique families is For infinite G, L f G = L G \ {M G }, and, for finite G, L f G = L G .
By abuse of notation, we write K(v) instead of K({v}), for v ∈ V . These particular clique families are building blocks for all other clique families: For a non-empty clique family K, every connected vertex subset C ∈ C G with K(C) = K is a generator of K. The set of generators of a clique family K is C(K). A generator C of K is minimal/maximal, iff it is so for set inclusion in C(K). There may be more than one minimal generator (see example 2.2). There is a unique maximal generator : In particular, for each non-empty clique family K, we have Proposition 2.1. Let K and K be clique families. Their sets of generators coincide, iff the clique families do so, and are disjoint otherwise.
Proof. We have the equivalence relation C ∼ C ⇔ K(C) = K(C ) on C G .
The clique families, their generators and maximal generators are: The clique family {K} has two minimal generators.
The clique families form a lattice with respect to set inclusion. All the chains in the lattice are finite and the lattice is both atomistic and co-atomistic. We use these facts later on, to reason inductively over this lattice.
Proposition 2.3. L G is a lattice with respect to set inclusion.
Proof. For K 1 , K 2 ∈ L G , define K 1 ∨ K 2 := K(C(K 1 ) ∩ C(K 2 )) , We claim that this is indeed the supremum and infimum of K 1 and K 2 in L G with respect to inclusion.
For the supremum property observe that each M ∈ K 1 contains C(K 1 ) and hence also C(K 1 ) ∩ C(K 2 ). Thus M must also be contained in K 1 ∨ K 2 . The same is true for every M ∈ K 2 , so K 1 ∨ K 2 is a common upper bound for K 1 and K 2 . To show that it is the least upper bound let K be an arbitrary upper bound. Then Hence, by the same argument as above, each M ∈ K 1 ∨ K 2 must be contained in K showing that K 1 ∨ K 2 ⊆ K.
A dual argument shows that the definition of the infimum is correct.
Note that K 1 ∧ K 2 = K 1 ∩ K 2 . In particular, L G is closed under intersections. Furthermore, the lattice has the following properties: • It is bounded with greatest element M G = K(∅) and smallest element ∅ = K(V ). By the remark following (4), all intervals not containing M G and, hence, all chains in L G are finite.
• L G is an atomistic lattice, the atoms being of the form Every clique family K = M G is the supremum of a finite set of atoms (see (6) and 7): • L G is a co-atomistic lattice, the co-atoms being of the form Each K ∈ L G is the infimum of finitely many co-atoms (see (5)): 3 Main result

Existence of clique trees of infinite chordal graphs
We investigate clique trees of infinite chordal graphs and extend the combinatorial construction of a subtree representation for finite graphs [Gav74]. A sensible definition of an infinite clique tree encompasses the fact that, for every induced subgraph, a corresponding induction on the cliques yields a clique tree of the induced subgraph. This gives a straightforward extension of the definition from the finite case.
Let G be infinite. A spanning tree T ∈ T M G is a clique tree of G, iff it fulfils (1), i.e. if every K(v) induces a tree.
Proposition 3.1. Every infinite and locally finite chordal graph has a clique tree.
The proof of proposition 3.1 is in section 6.1.

The characterisation via clique families
Let G be a (possibly infinite) chordal graph, let K ∈ L G , and denote by L < G (K) the strict subfamilies of K, i.e. the set Define a graph Γ K with vertex set K and an edge KL ∈ Γ K , iff there is Denote by ∼ K the equivalence relation whose classes are the connected components of Γ K and by [K] ∼ K the equivalence class of K with respect to the relation ∼ K .
Theorem 3.2. Let G be a locally finite chordal graph. A spanning subgraph T of M G is a clique tree of G, iff it fulfils one of the following equivalent conditions: T is a tree. In (9b), this fact is not so obvious, but follows from an inductive bottom-up construction over the lattice of clique families. The proof of theorem 3.2 is in section 6.2. Theorem 4.2 in the following sections shows that the conditions in (9b) are in fact non-overlapping.

Edge bijections and enumerating the set of clique trees
In the present section we take another look at the characterisation of clique trees via clique families in theorem 3.2 and disentangle the seemingly overlapping local conditions into disjoint local conditions. The key is differentiating between the restrictions imposed by a clique family and the restrictions imposed by its strict subfamilies. In this way, every restriction is dependent of and attached to a unique member of the lattice of clique families. We state this partition of the constraints in theorem 4.2 and apply it to counting and enumerating clique trees in the remainder of the section.
For K ∈ L G , define a graph Ξ K with vertex set K and an edge KL ∈ Ξ K , iff KL ∈ M G and K ∩ L ∈ C(K), equivalent to K(K ∩ L) = K. The graphs Γ K and Ξ K are edge-dual subgraphs of M G [K]: they have the same vertex set K and partition the edges of M G [K] into two disjoint sets.
Let ∆ K := Ξ K / ∼ K . The edges of ∆ K are injectively labelled by edges from Ξ K , as subgraph of M G .
Proposition 4.1. The graph ∆ K is complete, possibly with multi-edges and multi-loops. For each edge in ∆ K , its edge label KL fulfils K ∩ L = C(K), i.e. all the intersections of the edge labels coincide with the maximal generator.
Proof. If KL is not an edge of Ξ K , then K ∼ K L, whence they are identified in ∆ K . This shows the completeness of ∆ K .
By the definition of Ξ K , K ∩ L ∈ C(K) and thus K ∩ L ⊆ C(K). On the other hand, If G is infinite, then M G = K(∅) is the only infinite clique family and ∆ V consists of a single vertex and has no loops.
Theorem 4.2. There is a bijection between the edges of M G and the disjoint union over all clique families K of edges of Ξ K . Via edge-labelling, this extends to the disjoint union of edges of ∆ K .
Proof of theorem 4.2. Choose KL ∈ M G and K ∈ L f G . We know that KL ∈ Ξ K , iff K ∩ L ∈ C(K), equivalent to K(K ∩ L) = K. Proposition 2.1 and the identification between edges of Ξ K and edge-labels of ∆ K imply that we may partition the set of edges U according to the clique family: Theorem 4.2 tells us that condition (9b) factorises into a series of independent conditions. The characterisation (9b) of clique trees reduces the problem of choosing a clique tree to the problem of choosing a spanning tree of M G [K]/ ∼ K , for each K ∈ L G . Theorem 4.2 ensures that these choices are independent of each other. This is in contrast to characterisations (1) and (9a), where each edge might be subject to constraints from several clique families.
Corollary 4.3. Let G be a locally finite chordal graph. Then TC G , the set of clique trees of G, is in bijection with Proof. Using the bijection from theorem 4.2, we decompose the edges of a clique tree T ∈ TC G into disjoint sets, indexed by L f G . For K ∈ L f G , statement (9b) tell us that T K must be a spanning tree of ∆ K .
Conversely, select a spanning tree T K ∈ T ∆ K , for each K ∈ L f G , and let E be the union of their edge labels. By theorem 4.2, no edge in E appears twice as an edge-label of a T K . Let T be the subgraph of M G induced by E. By (9b) it is a clique tree.
A similar bijection to (11) between the clique trees of a finite chordal graph and a product of trees indexed by the minimal vertex separators (see section 5.3) of the graph is known [HL89]. The bijection is in fact the same, and we make the exact relation clear in corollary 5.10. An immediate consequence of (11) is a formula for the number of clique trees of a finite chordal graph: The value of |T ∆ K | is explicitely given in terms of the structure of ∆ K as a complete multigraph in [HL89]. The restriction amounts to a sequential processing of the clique trees.
Proof. As the degree is uniformly bounded, so is the size K and T ∆ K , for every K ∈ L f G . Furthermore, as each vertex is only contained in a uniformly bounded number of cliques and, hence, clique families, the size of L f G is linear in |V |. For infinite chordal graphs, we have a dichotomy in the number of clique trees: Corollary 4.5. Let G be an infinite chordal graph. It has either finitely or ℵ 1 many clique trees.
If a finite number of these numbers are greater than 1, then the number of clique trees is finite. If an unbounded number of these numbers are greater than 1, then there are at least a countable number of independent choices between more than two spanning trees and the number of clique trees is uncountable.

Classic properties of clique trees
We discuss classic properties of clique trees: the running intersection property, the maximal weight spanning tree property and the relation with minimal vertex separators and the reduced clique graph. We generalise several known results for finite graphs to the infinite graphs. For K ∈ L f G , let V (K) := {v ∈ K ∈ K} be the set of vertices covered by K. We start with a projection statement, which is our tool to lift properties from the finite to the infinite setting.

The maximal weight spanning tree property
Let w be the weight function on the edges of M G given by w(KL) := |K ∩ L|. Extend the weight function to T ∈ T M G , by setting w(T ) := KL∈T w(KL). Another classic characterisation of clique trees is: Condition (14) makes no sense in the infinite case. We can localise (14), though: Corollary 5.5. The tree T ∈ T M G is a clique tree, iff

Minimal separators and the reduced clique graphs
A non-empty subset W ⊆ V is a separator, iff G[V \ W ] has more than one connected component. It is a minimal separator, iff it is minimal with respect to inclusion.

Lemma 5.6 ([Dir61]). A (possibly infinite) graph is chordal, iff every minimal separator is complete.
Every minimal separator C separates two vertices adjacent to all of C. In particular, C is a minimal separator in G[V (K(C))].
The reduced clique graph [GHP95] R G of G is the subgraph of M G retaining those edges KL with K ∩ L a minimal separator and deleting the others. The importance of R G comes from: Lemma 5.7 ([GHP95]). Let G be a finite chordal graph. The union of all clique trees of G is R G .
Corollary 5.8. The union of clique trees of a chordal graph G is R G .
Proof. We apply the projection lemma 5.1 and the statement for the finite case in lemma 5.7, minding the remark after lemma 5.6.
The following lemma has been originally formulated only for finite graphs, but its proof is also valid in the infinite case: Lemma 5.9 ([HL89]). For T ∈ TC G , let C T be the multiset of intersections of edge-labels of T . The multiset C T is independent of T .
Corollary 5.10. A subset ∅ = W V is a minimal separator of G, iff W is the maximal generator of some clique family, i.e. W = C(K(W )). In particular, W must be complete and finite.
Proof. By corollary 5.8, every intersection of an edge label of a clique tree is a separator. By lemma 5.9, each minimal separator appears as intersection of at least one edge-label of every clique tree of T . By corollary 4.3 and proposition 4.1, the intersections of edge labels are exactly the maximal generators of finite clique families. 6 Proofs 6.1 Proof of existence of infinite clique trees We prove proposition 3.1 via a compactness argument, which is a rather standard approach in infinite graph theory (c.f. [Die05, Chapter 8.1]). Arguments of this type can often be used to obtain a result for infinite graphs from its finite counterpart.
Proof of proposition 3.1 by compactness. Let G be the graph. Let (v n ) n∈N be an enumeration of the vertices of G such that v n is connected to at least one v i for i < n. Denote by G n the subgraph of G induced by i≤n K∈K(vi) K, that is, G n contains all maximal cliques which contain at least one of v 1 , . . . , v n .
Since G n is a induced subgraph of a chordal graph it must be chordal as well. It is also connected. By construction every clique in G n corresponds to a clique in G, hence M Gn is a subgraph of M G . Since G n is finite, we know that we can find a clique tree T n of G n , that is, T n is a spanning tree of M Gn such that v → T n [K(v)] defines a subtree representation of G n .
Consider T n as a subgraph of M G and define a subgraph T of M G as follows. By local finiteness of G and thus M G , there is an infinite subsequence T 1 n of (T n ) n∈N of trees which contain the same edges of M G [K(v 1 )]. Add those edges to T . Then choose an infinite sub-subsequence T 2 n of T 1 n such that all elements of the sequence T 2 n contain the same edges of M G [K(v 2 )]. Proceed inductively.
We have to check that T is a tree and that T [K(v)] is a subtree, for every v ∈ V . The last property holds by construction. The trees corresponding to v and w overlap, iff K(v) ∩ K(w) = ∅, which is the case, iff vw is an edge. Hence T is connected because G was assumed to be connected. If T contains a cycle C, then it lies in M Gn 0 , for some n 0 . Hence C is a cycle in T n0 1 , a contradiction.

Proof of the clique family characterisation
Recall the definition of the strict subfamilies Γ K of a clique family K and the equivalence relation ∼ K from section 3.2. For ∅ = K ∈ Γ K and K ∈ K, we either have The major issue in the proof of theorem 3.2 is to start from (9b). In this case we build the tree bottom up, starting with the clique families {M }, for M ∈ M G . An important issue in later stages of the construction, for bigger clique families, is that overlapping constructions on strict subfamilies play well together. As the construction only adds edges, the main problem is not connectedness, but the possible introduction of cycles. Proposition 6.1 deals with this: for every connected component [K] ∼ K of Γ K , it asserts that there are no cycles introduced by the bottom up construction of the tree on smaller clique families.
The proof of proposition 6.1 is technical and is in section 6.2.2.
A second tool in the proof of theorem 3.2 is contracting and decontracting subtrees of trees. The following propositions, whose proofs are section 6.2.1 allow us to do the needed surgery on trees: Proposition 6.2. Let V be the vertex set of a finite graph and let V 1 , V 2 , . . . , V k be disjoint subsets of V . Every choice of two of the following statements implies the third one: is also a tree.
Equipped with these tools, we can prove our characterisation theorem: ] is a tree, for each v ∈ G, and that T [M G ] = T is a tree. This is just the definition of a clique tree.
(1) ⇒ (9a): is a tree. Let ∅ = K ∈ L G be arbitrary and assume that T [K] is not a tree. Assume that K is a maximal element of L G with the property that T [K] is not a tree. Such an element exists, because there are only finitely many elements of L G which are larger than K. Hence, if K is not maximal with this property, choose K K such that T [K ] is not a tree. Such a maximal family is neither empty (as T [∅] is a tree) nor induced by a single vertex. Let C be a minimal generator of K, i.e., C ⊆ C(K) and K(C) = K. The generator C contains at least two vertices. Therefore, for every ∅ = C C, Maximality of K implies that T [K(C )] and T [K(C \ C )] are trees. Proposition 6.3 implies that T [K] is a tree, too.
(9a) ⇒ (9b): let K ∈ L G . Proposition 6.1 together with the assumption that T [K ] is a tree for every K implies that T [[K] ∼ K ] is a tree, for every equivalence class with respect to the relation ∼ K . If K = M G , then there are only finitely many equivalence classes. Hence we can apply proposition 6.2 to show that T [K]/ ∼ K is a tree. For K = M G , the connectedness of G implies that there is only one equivalence class. Thus proposition 6.1 implies directly (without application of proposition 6.2) that T [K] is a tree.
(9b) ⇒ (9a): Assume that there is some K ∈ L G such that T [K] is not a tree. Choose K minimal with this property. This is possible because there are only finitely many elements of L G which are smaller than K. It follows from proposition 6.1 that T [[K] ∼ K ] is a tree for every equivalence class with respect to ∼ K . Since there are only finitely many equivalence classes and T [K]/ ∼ K is a tree we can invoke proposition 6.2 to prove that T [K] is indeed a tree, which completes the proof of the theorem.

Surgery on trees
This section contains technical results about the relation between subtrees obtained by inducing or contracting and the original tree and joining trees with common parts. The proofs of propositions 6.2 and 6.3 are also in this sections.
Proposition 6.4. Let V be the vertex set of a finite graph G and let W ⊆ V . Every choice of two of the following statements implies the third one: Proof. Denote by |G| and G the number of vertices and edges of a graph G respectively. If G is a tree, then holds. Contrarily, if (18) holds for a graph G and G is either acyclic or connected, then G is a tree.
For every graph G, the following identities hold: If two of the three statements in (17) hold, then (18) holds for them. Combined with (19), this yields (18) for the third statement of (17). Thus, in all three cases, we only need to show the acyclity or connectedness of the third graph.
(17c) and (17b) imply (17a): Every cycle in G is either contained in G[W ] or contracts to a cycle of G/W .
Proof of proposition 6.2. Apply proposition 6.4 inductively. The key fact is that G/{V 1 , . . . , V l }[V j ] is a tree, for all 1 ≤ l < j ≤ k.
Remark 6.5. Proposition 6.2 remains valid, if we consider locally finite graphs and countably many disjoint sets V i . It can also be extended to nested contractions, as long as the nesting depth is finite. If the nesting depth is infinite, then the limit object is no longer a spanning tree, but a topological spanning tree.
Proof of proposition 6.3. If either one of There is a unique path from v to v 1 and v 2 in T respectively. Thus, the edge v 1 v 2 can not be in T , as it would create a cycle in T . Hence, there are no edges between V 1 \ V 2 and V 2 \ V 1 in T . As T and T [V 2 ] are trees, proposition 6.4 implies that T /V 2 = T [V 1 ]/V 1 ∩ V 2 is a tree, too. As T [V 1 ] is a tree, another application of proposition 6.4 implies that Proposition 6.6. Let G be a graph with vertices V . Let V =: are trees, and that there are no edges connecting V 1 \ V 2 to V 2 \ V 1 . Then G is a tree.
Proof. As G[V 1 ] and G[V 1 ∩ V 2 ] are trees, we apply proposition 6.4 to see that As G/V 2 and G[V 2 ] are trees, another application of proposition 6.4 yields that G is also a tree.
6.2.2 The proof of proposition 6.1 First, we establish an additional property of cycles in chordal graphs, needed in the proof of proposition 6.1.
Proposition 6.7. If G is a chordal graph, then every cycle of length ≥ 4 in G contains a 2-chord, i.e. a chord connecting two vertices with distance 2 along the cycle.
Proof. Let C be a cycle of G of length k ≥ 4. As G is chordal, it has a chord e 1 which splits it up into two cycles. If one of those two cycles has length 3 (including the chord), then we are done. Otherwise, take one of the cycles, C 1 , and split it along a chord e 2 into two cycles. Denote by C 2 the cycle from the C 1 -splitting not containing e 1 . Its only non-C edge is e 2 . If C 2 has length 3, then we are done. Otherwise proceed recursively, with each C i having e i as its only non-C edge. Since the lengths of the cycles C i are strictly decreasing, the recursion terminates.
Proof of proposition 6.1. T is connected: If K ∼ K K, then K is connected to K by a path in Γ K . Hence, we can find a sequence K 1 , K 2 , . . . , K k of families in L < G (K) such that K ∈ K 1 , K ∈ K k and K i ∩ K i+1 = ∅, for 1 ≤ i < k. As K ∈ K 1 , K 1 is completely contained in [K] ∼ K . Since K i and K i+1 have nonempty intersection, it follows by induction that each K i contains some element of [K] ∼ K and is also completely contained in [K] ∼ K . As T [K i ] is a tree, for every i, it is connected. We choose and combine paths from T [K i ] to obtain a path from K to K in T .
T is acyclic: Assume that there is a cycle C := K 1 K 2 . . . K n in T . Since K n and K 1 are connected by an edge in Γ K there must be some K ∈ L < G (K) which contains both of them, that is K(K n ∩ K 1 ) ⊆ K K. Since this K has nonempty intersection with [K] ∼ K it is completely contained in [K] ∼ K .
Define the indices i 1 , . . . , i r inductively by: If i j < n, then define As 1 ≤ i 1 < i 2 < . . ., this construction stops after r ≤ n steps with i r = n. Let . . .
By construction, it holds that K j : We choose C and its cyclic ordering minimizing the value of r.
Case r = 1: The tree T [K 1 ] contains the cycle C, a contradiction.
Case r = 2: By the minimality of r, we have {K i1 , K i2 } ⊆ K 1 ∩ K 2 =: ] are all trees we apply proposition 6.6 to T [K 1 ] ∪ T [K 2 ] and deduce that it is a tree and does not contain the cycle C.
Case r = 3: We claim that Admitting the claim for the moment, we can finish the proof of the case r = 3. We apply proposition 6.3 three times: First, to the trees Third, to T [K 1 ] ∪ T [K 2 ], T [K 3 ] and T [K 1 ∩ K 3 ] ∪ T [K 2 ∩ K 3 ]. We check that the third tree is indeed the intersection of the first two. For the vertex sets, we have (K 1 ∪ K 2 ) ∩ K 3 = (K 1 ∩ K 3 ) ∪ (K 2 ∩ K 3 ) .
Proof of the claim (20): Since C 1 ∪ C 2 ∈ K i1 , C 1 ∪ C 3 ∈ K i3 and C 2 ∪ C 3 ∈ K i2 , they all induce complete subgraphs of G. For a collection of subsets {C i } i∈[n] of V , we have: Hence, C 1 ∪ C 2 ∪ C 3 is a complete subset of G and there is a clique M ∈ M G containing C 1 ∪ C 2 ∪ C 3 , establishing (20).
Case r ≥ 4: We construct a cycle in G violating proposition 6.7. Our first observation is To prove this observation, we assume that 1 ≤ i 1 < i 2 ≤ r and D := C i1 ∪C i2 . If i 2 − i 1 = 1, then D ∈ M i1 . Hence D is complete. If i 2 − i 1 ≥ 2 and D = ∅, then there is a clique M containing D with M ∈ K i1 ∩ K i2 . This allows the creation of a cycle C , by shortcutting C via a connection from M i1 to M in T [K i1 ] and from M to M i2−1 in T [K i2 ]. We can cover the cycle C by at most i 2 − i 1 + 1 < r clique families, contradicting the minimality of r.
where the maximum runs over a non-empty set, since by (22) v 3 is adjacent to each vertex in C 4 . Every vertex in C j4 is adjacent to v 3 , but, for every 4 ≤ j 4 < j ≤ r, there is a vertex in C j not adjacent to v 3 . Because of observation (22), we can choose non-adjacent vertices v 2 ∈ C 2 and v 4 ∈ C j4 . The path v 1 v 2 v 3 v 4 is two-chordless (23), for l = 4.
We extend this two-chordless path until we end up with a cycle contradicting proposition 6.7. Because the path always fulfils (23d), it has at most length r.
Let v l ∈ C j l be the last element of the path. We have two cases: Case v l is adjacent to v 1 : We can complete the v 1 . . . v l to a cycle. It fulfils (23a) and (23b), contradicting proposition 6.7.
Case v l is not adjacent to v 1 : We extend our path. This happens at most r − 4 times. By (22), we know that j l < r. Hence, let j l+1 := max{j ≤ r | C j ∪ {v l } ∈ C G , } .