Toughness and Hamiltonicity in Random Apollonian Networks Lilian Markenzon ∗ NCE - Universidade Federal do Rio de Janeiro [email protected] Christina F. E. M. Waga IME - Universidade do Estado do Rio de Janeiro [email protected] Abstract In this paper we study the toughness of Random Apollonian Net- works (RANs), a random graph model which generates planar graphs with power-law properties. We consider their important characteristics: every RAN is a uniquely representable chordal graph and a planar 3-tree and as so, known results about these classes can be particularized. We establish a partition of the class in eight nontrivial subclasses and for each one of these subclasses we provide bounds for the toughness of their elements. We also study the hamiltonicity of the elements of these subclasses. Keywords: randon Apollonian network, planar k-tree, clique-tree, tough- ness, hamiltonicity 1 Introduction Over the last few years, the ever growing interest in social networks, the Web arXiv:2002.09244v1 [math.CO] 21 Feb 2020 graph, biological networks, etc., led to a great deal of research being built around modelling real world networks. In 2005, Andrade et al. [1] introduced Apollo- nian networks (AN s), inspired by Apollonian packings [12], that proved to be an interesting tool for modeling real networked systems. These networks can be produced as follows: start with a triangle and then at each iteration, inside each triangle, a vertex is added and linked to the three vertices. Apollonian networks are scale-free, display the small-world effect and have a power-law de- gree distribution. Generalizing ANs, the Random Apollonian Networks (RAN s) were introduced by Zhou et al. [22]; in this case, at each iteration of a RAN a ∗Partially supported by grant 304706/2017-5, CNPq, Brazil. triangle is randomly selected. Some problems have been solved for these classes. For instance, an exact analytical expression for the number of spanning trees in ANs was achieved by Zhang et al. [21]; the degree distribution, k largest degrees and k largest eigenvalues (for a fixed k) and the diameter of RANs were studied in Frieze and Tsourakakis [8]; Ebrahimzadeh et al. [7] follow this line of research by studying the asymptotic properties of the longest paths and pre- senting sharp estimates for the diameter of a RAN. Others papers had employed a non-deterministic concept. In this paper we focus in a different approach. Considering the equivalence between RAN s and the planar 3-trees ( i.e., the maximal chordal planar graphs [15]) we analyse vulnerability properties of the networks, based on their clique- trees. The toughness of a graph is an important invariant introduced in 1973 by Chv´atal [6] that deals with the vulnerability of a graph. Let the number of components of a graph G = (V, E) be denoted by ω(G). A graph G is t-tough if |S|≥ tω(G − S) for every subset S ⊆ V with ω(G − S) > 1. The toughness of G, denoted τ(G), is the maximum value of t for which G is t-tough (taking τ(Kn)= ∞, n ≥ 1). In other words, the toughness relates the size of a separa- tor with the number of components obtained after deleting it. It is important to highlight that the toughness can be directly related to the hamiltonicity of the graph. Chv´atal [6] has established that every Hamiltonian graph is 1-tough, but 1-toughness does not ensure hamiltonicity. He has also conjectured that there exists a t such that every t-tough graph is Hamiltonian. Some papers prove Chv´atal’s conjecture for different graph classes: τ(G) ≥ 3/2 for a split graph [13], τ(G) > 1 for planar chordal graphs [4], τ(G) ≥ 3/2 for spider graphs [11] and τ(G) ≥ 1 for strictly chordal graphs [18]. In particular for k-trees, Broersma et al. [5] presented important results, showing that if G is a k-tree, k ≥ 2, with toughness at least (k + 1)/3, then G is Hamiltonian. For k = 2, they prove that every 1-tough 2-tree on at least three vertices is Hamiltonian. Kabela [10] has improved this result, showing that every k-tree (except for K2) with toughness greather than k/3 is Hamiltonian. In this paper we study the toughness of Random Apollonian Networks based on their characteristics: every RAN is a uniquely representable chordal graph and, as so, it has a unique clique-tree; every RAN is a planar 3-tree and the results of B¨ohme et al. [4] and Broersma et al. [5] can be particularized. We establish a partition of the class in eight nontrivial subclasses reliant on the structure of the clique-tree, and for each one of these subclasses we provide bounds for the toughness of their elements. We also study the hamiltonicity of the elements of these subclasses. Some well-known graphs, as the Goldner-Harary graph and the Nishizeki’s example of a non-Hamiltonian maximal planar graph [19], fall in one of the defined subclasses. 2 2 Background Let G = (V, E), be a connected graph, where |V | = n and |E| = m. The set of neighbors of a vertex v ∈ V is denoted by N(v) = {w ∈ V ; {v, w} ∈ E}. The degree of a vertex v ∈ V is d(v) = |N(v)|. For any S ⊆ V , the subgraph of G induced by S is denoted G[S]. If G[S] is a complete graph then S is a clique in G. A vertex v ∈ V is said to be simplicial in G when N(v) is a clique in G. The set of simplicial vertices of G is denoted by SI. The graphs G = (V, E) and G′ = (V ′, E′) are isomorphic if there is a bijective function f : V → V ′ such that for all v, w ∈ V, {v, w} ∈ E if and only if {f(v),f(w)} ∈ E′, i.e, f preserves adjacency. Basic concepts about chordal graphs are assumed to be known and can be found in Blair and Peyton [3] and Golumbic [9]. In this section, the most pertinent concepts are reviewed. 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]. Let G = (V, E) be a chordal graph and u, v ∈ V . 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. When the pair of vertices remains unspecified, we refer to S as a minimal vertex separator (mvs). The set of minimal vertex separators is denoted by S. The clique-intersection graph of a graph 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. Each edge is assigned an inte- ger weight, given by the cardinality of the intersection between the maximal cliques represented by its endpoints. Every maximum-weight spanning tree of the clique-intersection graph of G is called a clique-tree of G. The set of max- imal cliques of G is denoted by Q. A clique-tree of G represents the graph G. Clique-trees satisfy the induced subtree property (ISP): Q(v) induces a subtree of the clique-tree T of G where Q(v) is the set of maximal cliques containing the vertex v ∈ V . Observe that each maximal clique Q ∈ Q is related to a vertex q of the clique-tree T of G. A simplicial clique is a maximal clique containing at least one simplicial vertex. For a chordal graph G and a clique-tree T of G, a set S ⊂ V is a mvs of G if and only if S = Q ∩ Q′ for some edge {Q,Q′} in T . Moreover, the multiset M of the minimal vertex separators of G is the same for every clique-tree of G. The multiplicity of the minimal vertex separator S, denoted by µ(S), is the number of times that S appears in M. The determination of the minimal vertex separators and their multiplicities can be performed in linear time [16]. A k-regular tree is a tree in which every vertex that is not a leaf has degree k. 3 3 Some subclasses of chordal graphs In this paper we deal with some subclasses of chordal graphs which are now reviewed. A chordal graph is called a uniquely representable chordal graph [14] (briefly ur-chordal graph) if it has exactly one clique-tree. Theorem 1 [14] Let G be a chordal graph. Then, G is uniquely representable if and only if there is no proper containment between any minimal vertex sepa- rators and all minimal vertex separators are of multiplicity one. A k-tree, k> 0, firstly presented in [20], can be inductively defined as follows: 1. Every complete graph with k + 1 vertices is a k-tree. 2. If G = (V, E) is a k-tree, v∈ / V and S ⊆ V is a k-clique of G, then G′ = (V ∪{v}, E ∪{{v, w} | w ∈ S}) is also a k-tree. 3. Nothing else is a k-tree. Two subclasses of k-trees are the simple-clique k-trees (SC k-trees) and the k- path graphs [15]. A SC k-tree, k > 0, is a uniquely representable k-tree. A complete graph on k + 1 vertices is a k-path graph, k > 0; if n > k + 1, G is a k-path graph if and only if G has exactly two simplicial vertices.