New Results on Ptolemaic Graphs

L. Markenzon NCE - Universidade Federal do Rio de Janeiro [email protected] C. F. E. M. Waga IME - Universidade do Estado do Rio de Janeiro [email protected] M. C. Golumbic xxxx

August 31, 2013

Abstract In this paper, we analyze ptolemaic graphs for its properties as chordal graphs. Firstly, two characterizations of ptolemaic graphs are proved. The first one is based on the reduced clique graph, a structure that was defined by Habib and Stacho [8]. In the second one, we simplify the characterization presented by Uehara and Uno [13] with a new proof. Then, known subclasses of ptolemaic graphs are reviewed in terms of minimal vertex separators. We also define another subclass, the laminar chordal graphs, and we show that a hierarchy of ptolemaic graphs can be built based on characteristics of the minimal vertex separators in each subclass.

Keywords: ptolemaic graph, minimal vertex separator, block duplicate graph, AC graph, laminar chordal graph.

1 Introduction

Ptolemaic graphs have been studied for some time. However, their basic properties as chordal graphs are not yet fully explored. In this paper, we analyze ptolemaic graphs for its properties as chordal graphs. Firstly, two characterizations of ptolemaic graphs are proved. The ﬁrst one is based on the reduced clique graph, a structure that was deﬁned by Habib and Stacho [8]. In the second one, we simplify the characterization presented by Uehara and Uno [13]

1 with a new proof, analyzing the behavior of the one-vertex extensions [1]. Then, the block duplicate graphs, a subclass of ptolemaic graphs, are reviewed and a new characterization that leads to a simple recognition algorithm is presented. We also define another subclass, the laminar chordal graphs, and we show that these graphs are double-diamond-free. A new hierarchy of ptolemaic graphs can be built based on characteristics of the minimal vertex separators in each subclass. Basic concepts about chordal graphs are assumed to be known and can be found in Blair and Peyton [3] and Golumbic [6]. Let G = (V,E) be a connected graph, being |E| = m its size and |V | = n its order. The set of neighbors of a vertex v ∈ V is denoted by N(v) = {w ∈ V | {v, w} ∈ E} and its closed neighborhood by N[v] = N(v) ∪ {v}. Two vertices u and v are true twins in G if N[u] = N[v] and they are false twins in G if N(u) = N(v). For any S ⊆ V , let G[S] be the subgraph of G induced by S. If G[S] is complete then S is a clique. A vertex v is said to be simplicial in G when N(v) is a clique in G. Let G = (V,E) be a graph and u, v ∈ V . A subset S ⊂ V is a separator of G if at least two vertices in the same connected component of G are in two distinct connected components of G[V −S]. The set S is a minimal separator of G if S is a separator and no proper subset of S separates the graph. A subset S ⊂ V is a vertex separator for non-adjacent vertices u and v (a uv-separator) if the removal of S from the graph separates u and v into distinct connected components. If no proper subset of S is a uv-separator then S is a minimal uv- separator. If S is a minimal uv-separator for some pair of vertices, it is called a minimal vertex separator (mvs). An efficient algorithm to determine the set of minimal vertex separators can be find in [12]. Dirac [5] has proved an important characterization of chordal graphs.

Theorem 1 [5] G is chordal if and only if every minimal vertex separator of G is a clique.

The set of maximal cliques of G is denoted Q. The clique-intersection graph of a chordal graph G, C(G), is the connected weighted graph whose vertices are the maximal cliques of G and whose edges connect vertices corresponding to non-disjoint maximal cliques. A clique-tree of G is defined as a tree T whose vertices are the maximal cliques of G such that for every two maximal cliques Q and Q0 each clique in the path from Q to Q0 in T contains Q ∩ Q0. Blair and Peyton [3] proved that, for a chordal graph G = (V,E) and a clique- tree T = (VT ,ET ), a set S ⊂ V is a minimal vertex separator of G if and only if 0 00 0 00 S = Q ∩ Q for some edge {Q ,Q } ∈ ET . They also proved that every clique- tree T of G satisfies the induced-subtree property: for every vertex v ∈ V , the set Qv (which denotes the set of maximal cliques containing the vertex v) induces a subtree of T . Moreover, the multiset M of the minimal vertex separators of G is the same for every clique-tree of G. It is clear that |M| = q − 1, q being the number of maximal cliques of G. We define the multiplicity of the minimal

2 vertex separator S, denoted by µ(S), as the number of times that S appears in M. The set of minimal vertex separators of G is denoted S. The diamond, gem, dart and double-diamond are the graphs illustrated in Figure 1. A graph is gem (diamond, dart, double-diamond)-free when it does not contains the gem (diamond, dart, double-diamond) as an induced subgraph. For example, a block graph is a diamond-free chordal graph.

Figure 1: Gem, diamond, dart and double-diamond graphs.

2 Ptolemaic graphs - basic concepts

In this section, basic concepts of ptolemaic graphs are reviewed and we establish the relation between these concepts and structural properties of chordal graphs as maximal cliques and minimal vertex separators. A connected graph G is distance hereditary if the distance between any two vertices remains the same in every connected induced subgraph. A connected graph G is ptolemaic if for any four vertices u, v, w, x of V ,

d(u, v)d(w, x) ≤ d(u, w)d(v, x) + d(u, x)d(v, w).

Theorem 2 [9] The following conditions are equivalent:

1. G is ptolemaic. 2. G is gem-free and chordal. 3. G is distance hereditary and chordal. 4. For all distinct non-disjoint maximal cliques Q and Q0 of G, Q ∩ Q0 separates Q\Q0 and Q0\Q.

Bandelt and Mulder [1] presented inductive constructions of distance hereditary and ptolemaic graphs. Let G = (V,E) be a graph, v ∈ V and u∈ / V . An extension of G to a graph G0 = (V 0,E0) is a one-vertex extension if it obeys one of the following three rules:

(α) V 0 = V ∪ {u} and E0 = E ∪ {v, u}, i.e., a pendant vertex u is added.

3 (β) V 0 = V ∪ {u} and E0 = E ∪ {{x, u} | x ∈ N[v]}, i.e., u is a true twin of v. (γ) V 0 = V ∪ {u} and E0 = E ∪ {{x, u} | x ∈ N(v)}, i.e., u is a false twin of v.

They have shown that a distance hereditary graph G is obtained from a complete graph with two vertices by a sequence of one-vertex extensions.

Figure 2: Ptolemaic graph.

Theorem 3 [1] A graph G is ptolemaic if and only if G is obtained from a single vertex by a sequence of one-vertex extensions where applications of (γ) are restricted to vertices x whose (non-empty) neighborhoods induce complete subgraphs.

The special case of (γ) deﬁned in Theorem 3 will be denoted by (γ∗) and included in the set of rules to build a one-vertex extension of a graph:

(γ∗) V 0 = V ∪ {u} and E0 = E ∪ {{x, u} | x ∈ N(v) 6= ∅, v simplicial}, i.e., u is a false twin of v.

In order to generate a ptolemaic graph G = (V,E), it is possible to establish a building sequence. A ptolemaic extension sequence (pes) of G is a sequence of triples Π(G) = [π(1), . . . , π(n)] being π(i) = (ei, vi, ui), i = 2, . . . , n, such that

∗ ∗ 1. ei ∈ {α, β, γ }, being e2 6= γ ;

2. vi = uj, for some j < i;

3. ui 6= uk, 1 ≤ k ≤ i − 1; and π(1) is the special initial triple ( , , u1) where the symbol denotes the absence of the parameter.

4 Some subclasses can be easily described. Consider a pes Π(G) of a ptolemaic graph G. If it is composed only by type (α) or type (γ∗) extensions, the graph G is a tree and if it is composed of only type (β) extensions, the graph G is a complete graph.

3 Characterization of ptolemaic graphs

3.1 Reduced clique graphs

The following notions were presented by Habib and Stacho [8] in order to propose a new representation for chordal graphs, the reduced clique graph, that was used to characterize asteroidal sets in chordal graphs and to discuss chordal graphs that admit a tree representation with small number of leaves. We now show that this representation has an special structure when the graph is ptolemaic.

Two maximal cliques Q and Q0 of G form a separating pair if Q ∩ Q0 is non- empty and every path in G from a vertex of Q\Q0 to a vertex of Q0\Q contains a vertex of Q ∩ Q0.

Theorem 4 [8] A set S is a minimal vertex separator of a chordal graph G if and only if there exist maximal cliques Q and Q0 of G forming a separating pair such that S = Q ∩ Q0.

The reduced clique graph Cr(G) of G is the graph whose vertices are maximal cliques of G and whose edges {Q, Q0} are between cliques Q and Q0 forming separating pairs.

Theorem 5 [8] Let G be a connected chordal graph. The reduced clique graph Cr(G) is the union of all clique-trees of G.

The reduced clique graph and the concept of separating pair provide a new characterization for ptolemaic graphs.

Theorem 6 Let G = (V,E) be a chordal graph. The following statements are equivalent:

1. G is ptolemaic. 2. Every pair of distinct non-disjoint maximal cliques Q and Q0 of G form a separating pair. 3. The reduced clique graph and the clique-intersection graph are the same.

Proof.

5 (1 → 2) Consider G a ptolemaic graph. By Theorem 2, for all distinct non- disjoint maximal cliques Q and Q0 of G, Q ∩ Q0 separates Q\Q0 and Q0\Q. Then, every path in G from a vertex of Q\Q0 to a vertex of Q0\Q contains a vertex of Q ∩ Q0. So, Q and Q0 form a separating pair.

(2 → 3) Let C(G) = (Q,Ec) be the clique-intersection graph and Cr(G) = (Q,Er) be the reduced clique graph of a ptolemaic graph G. By Theorem 5, 0 Er ⊆ Ec. Consider an edge {Q, Q } ∈ Ec. All pairs of distinct non-disjoint maximal cliques Q and Q0 of G form a separating pair. Then, by Theorem 4, Q ∩ Q0 is a minimal vertex separator of G. So, the edge {Q, Q0} belongs to at least one clique-tree of G. Then, Ec ⊆ Er.

(3 → 1) Consider C(G) = Cr(G). By the deﬁnition of reduced clique graph, all intersections of maximal cliques are minimal vertex separators of G, i.e., all pairs of distinct non-disjoint maximal cliques are separated by their intersection. Then, by Theorem 2, G is ptolemaic.

3.2 Laminar sets

Uehara and Uno introduced in [13] a representation for ptolemaic graphs called clique laminar tree and they proved the laminarity of particular cliques of the graph. We show here that in order to characterize ptolemaic graphs only the minimal vertex separators need to be considered. Our proof is diﬀerent from the one presented by them.

Firstly, some theoretical results must be presented. Let F = {F1, ..., Fk},Fi ⊂ V, 1 ≤ i ≤ k, be a family of sets. F is a laminar family if Fi ∩ Fj 6= ∅ implies that Fi ⊆ Fj or Fj ⊆ Fi, 1 ≤ i, j ≤ k. In other words, a family of sets is called laminar if any two of the sets are disjoint or one of them is subset of the other. An empty family is a laminar family.

Theorem 7 Let G = (V,E) be a chordal graph. G is ptolemaic if and only if the family of minimal vertex separators contained in each maximal clique of the graph is laminar.

Proof. (The proof is by induction.)

The ptolemaic graph G = ({u1}, ∅) has one maximal clique and the laminarity of the family of minimal vertex separators is trivially satisﬁed. Let G = (V,E) be a ptolemaic graph with |V | = n − 1 such that the family of minimal vertex separators contained in each maximal clique of the graph is laminar and let G0 = (V 0,E0) be a one-vertex extension of G applying the rules (α), (β) and (γ∗), v ∈ V , u∈ / V and V 0 = V ∪ {u}. Three cases may occur:

• (α, v, u) In this case, E0 = E ∪ {{v, u}} and G0 has one more maximal clique Q than G separated by the mvs {v}. If v is a simplicial vertex in G, {v}

6 is not a subset of another mvs and the laminarity of the family of the minimal vertex separators in Q is maintained; otherwise, v ∈ S, {v} ⊆ S and the laminarity is also maintained. • (β, v, u) Vertex u is a true twin of v in G0. If v is a simplicial vertex in G, the maximal clique to which it belongs is enlarged by including the vertex u, the minimal vertex separators of the graph are not modified and the laminarity is preserved; otherwise, all maximal cliques and all minimal vertex separators to which it belongs are enlarged by including the vertex u. So, the containment property is preserved and consequently also the laminarity property. • (γ∗, v, u) By definition, v is a simplicial vertex. If v is a pendant vertex, (γ∗, v, u) is equivalent to (α, N(v), u). Consider v a non-pendant vertex and Q the maximal clique to which it belongs. Vertices u and v are false twins in G0. The graph G0 has one more maximal clique Q0 = N(v) ∪ {u} than G. If N(v) is already a mvs of G, the family of minimal vertex separators is the same in both graphs and the laminarity is maintained. If not, N(v) is a new mvs and two families of minimal vertex separators must be considered, associated to Q0 and Q. The first one is a unitary family, so it is laminar. The family contained in Q is laminar in G, by hypothesis. When this family is considered in G0, it has a new element N(v). As v is a simplicial vertex, for all mvs S of the family contained in Q, S ⊂ N(v). So the laminarity is maintained. Consequently, if G is a ptolemaic graph then the family of minimal vertex separators contained in each maximal clique of the graph is laminar.

Conversely, let us consider a non-ptolemaic (but chordal) graph G. By Theorem 2, G contains a gem graph as an induced subgraph, then there exist two maximal cliques Q0 and Q00 such that Q0 ∩ Q00 does not separate Q0 and Q00, v ∈ Q0 ∩ Q00, u ∈ Q0\Q00 and w ∈ Q00\Q0 in such a way that {u, v, w} is a clique belonging to a maximal clique Q. Assume without loss of generality that Q = {u, v, w}. So, {u, v} ⊆ Q0 ∩ Q and {v, w} ⊆ Q00 ∩ Q are minimal vertex separators. Then, the family of minimal vertex separators contained in the maximal clique Q is not laminar.

4 Known subclasses

In this section, two known subclasses of ptolemaic graphs are reviewed in terms of minimal vertex separators: the block duplicate graphs and the AC graphs.

7 The block duplicate graphs were introduced by Golumbic and Peled [7]. The class was also deﬁned as strictly chordal graphs in [10] based on hypergraph properties. The class was proved to be gem-free and dart-free ([7], [11]). A block duplicate graph is a graph obtained by adding zero or more true twins to each vertex of a block graph G. Block duplicate graphs also have properties in terms of the structure of their minimal vertex separators as proved in the following theorem. Based on the theorem, the recognition algorithm for block duplicate graphs becomes trivial.

Theorem 8 Let G = (V,E) be a chordal graph and S be the set of minimal vertex separators of G. The following statements are equivalent: 1. G is a block duplicate graph.

2. For any distinct S, S0 ∈ S, S ∩ S0 = ∅. 3. G is gem-free and dart-free.

Proof. (1 → 2) Let G be a block duplicate graph built from a block graph G0. All the minimal vertex separators of a block graph have cardinality 1. By the deﬁnition of block duplicate graph, only true twins can be added to G0. If a new vertex v is twin of a simplicial vertex w, the maximal clique that contains w is augmented and the new vertex v is also simplicial in G. Otherwise, w belongs to a minimal vertex separator S and S has its cardinality augmented; only this mvs and the maximal cliques to which it belongs are modiﬁed. So, the intersection between distinct separators of G remain empty. (2 → 3) The gem and the dart graphs have three maximal cliques and two distinct minimal vertex separators S and S0, being, in both, S ∩ S0 6= ∅. (3 → 1) By Theorem 1 of [7].

Another subclass of ptolemaic graphs, the AC graphs were presented in [2]. An AC graph is a graph whose clique-intersection graph is acyclic. Their characterization is simple and they are easy to recognize.

Theorem 9 [4] A graph G = (V,E) is an AC graph if and only if it is chordal and every vertex in G belongs to at most two cliques.

Observe that it is not possible to establish a containment relation between block graphs and AC graphs. Figure 3 shows in (a) an example of a block graph which is not an AC graph and in (b) an AC graph that is not a block graph. Moreover, the block duplicate graphs generalize both classes. The next theorem shows the behavior of the minimal vertex separators and their multiplicity (µ) for AC graphs. The proof is immediate.

Theorem 10 A chordal graph G is an AC graph if and only if for any distinct S, S0 ∈ S, S ∩ S0 = ∅ and for all S ∈ S, µ(S) = 1.

8 Figure 3: Ptolemaic graphs.

5 Laminar chordal graphs

In extending the notions stated in Theorem 7, we deﬁne a new class which takes into account the laminarity of the whole family of minimal vertex separators, resulting in a subclass of ptolemaic graphs. Consequently, a new hierarchy of ptolemaic graphs can be built based on characteristics of their minimal vertex separators.

A chordal graph G is called a laminar chordal graph if the set S of all minimal vertex separators is laminar. This class can be characterized as follows.

Theorem 11 Let G = (V,E) be a chordal graph. G is laminar chordal graph if and only if G is gem-free and double-diamond-free.

Proof. Neither the family of minimal vertex separators of the gem graph nor the family of the double-diamond graph are laminar. Conversely, let us consider a non-laminar chordal graph G. Then, there are two distinct minimal vertex separators S and S0 of G such that S ∩ S0 6= ∅, S 6⊆ S0 and S0 6⊆ S. Assume without loss of generality that S = {u, v} and S0 = {v, w} with u, v, w ∈ V . Two cases must be studied. There is a maximal clique Q ∈ Q such that S ∪ S0 ⊆ Q. So, {u, v, w} ⊆ Q and there are maximal cliques Q1,Q2 ∈ Q such that {u, v} ⊂ Q1 and {v, w} ⊂ Q2. For any clique-tree T of G, the tree of the Figure 4 is a subtree. Then, G contains the gem graph as a induced subgraph and, by Theorem 2, it is not a ptolemaic graph.

{u, v} {v, w} Q1 Q = {u, v, w, . . .} Q2

Figure 4: Subtree of a clique-tree T .

Otherwise, for every maximal clique Q of G, S ∪ S0 6⊆ Q. Thus, there are maximal cliques Q1,Q2,Q3,Q4 ∈ Q such that Q1 ∩ Q2 = Q1 ∩ Q4 = Q2 ∩ Q3 = 0 Q2 ∩ Q4 = S ∩ S = {v}. By the induced-subtree property of chordal graphs,

9 {u, v} Q1 Q2

{v, w} Q3 Q4

Figure 5: Subgraph of the clique-intersection graph. for every clique-tree, only one of the dotted edges in Figure 5 has to be chosen. Then, G contains the double-diamond graph as a induced subgraph. Any other possibility is a variation from these two cases.

Knowing that a laminar chordal graph is a ptolemaic graph, a characterization is obtained by using an inductive construction based on the one-vertex extensions applying only the rules (α), (β) and (γ∗).

Theorem 12 Let G = (V,E) be a ptolemaic graph. G is laminar chordal graph if and only if G is obtained from a single vertex by a ptolemaic extension sequence where the application of (γ∗) is restricted to a vertex v such that S Si ⊆ N(v) where v = {Si ∈ | Si ∩ N(v) 6= ∅}. Si∈Sv S S

Proof. The analysis of the application of (α) or (β) is similar to the one presented in Theorem 7 and both maintain the laminarity of S. We must further analyze the eﬀect of the application of (γ∗) on S. If N(v) ∈ S, the set of minimal vertex separators does not change. Let us analyze the case N(v) ∈/ S. Other simplicial vertices can belong to N(v). The set of non-simplicial vertices of N(v) is denoted N 0(v). It is clear that N 0(v) = 0 0 0 0 0 S1 ∪ S2 ∪ ... ∪ Sk, k ≥ 1 and Si ∈ S, 1 ≤ i ≤ k, that is, N (v) is composed by one or more minimal vertex separators. After (γ∗, v, u) there is a new mvs N(v). From these considerations, the following facts are equivalent: S S 6⊆ N(v); some of the minimal vertex Si∈ v i 0 0 0 S separators S1,S2,...,Sk are properly contained in a minimal vertex separator Sj ∈ S, Sj ∩ N(v) 6= ∅; and the laminarity in not maintained. This concludes the proof. In terms of algorithms, it is important to state that the recognition algorithm for the class is linear and follows directly from the deﬁnition. Theorem 12 provides us an interative method of construction. Finally, Figure 6 presents the chain of inclusion relations among ptolemaic graphs and their subclasses.

10 ptolemaic

laminar chordal

block duplicate

block graph AC tree

path

Figure 6: A hierarchy of ptolemaic graphs.

References

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