Characterizing Block Graphs in Terms of One-vertex Extensions

Block graphs has been extensively studied for many decades. In this paper we present a new characterization of the class in terms of one-vertex extensions. To this purpose, a specific representation based on the concept of boundary cliques is presented, bringing about some properties of the graph.


INTRODUCTION
Block graphs have been extensively studied for many decades with characterizations based on different approaches since the first one in 1963 until today. In this paper we present a new characterization of the class in terms of one-vertex extensions. As block graphs are a subclass of chordal graphs, properties of this class can be successfully particularized: a specific representation of block graphs based on the concept of boundary cliques is presented, bringing about some properties of the graph.
Harary [7] introduced the definition of a block graph based on structural properties and presented a classical characterization: the block graph B(G) of a given graph G is that graph whose vertices are the blocks (maximal 2-connected components) B 1 , . . . , B k of G and whose edges are determined by taking two vertices B i and B j as adjacent if and only if they contain a cut-vertex (its removal disconnects the graph) of G in common. A graph is called a block graph if it is the block graph of some graph. 1. the shortest path between any two vertices of G is unique and 2. for each edge e = uv ∈ E, if x ∈ N e (u) and y ∈ N e (v), then, and only then, the shortest path between x and y contains the edge e, where N e=uv Bandelt and Mulder [1] presented a characterization based on forbidden subgraphs.

Characterization 5. [1]
A graph is a block graph if and only if it is C n≥4 -free and diamond-free.
Mulder and Nebeský [11] characterized block graphs using an algebraic approach, a binary operation + (leap operation) on a finite nonempty set V such that for u, v, w ∈ V , The underlying graph G + = (V, E + ) of + is such that uv ∈ E + if and only if u = v, u + v = v and v + u = u.
Intuitively, for any two vertices u and w in different blocks, the leap operation produces the cutvertex z in the block of u on the way to w, i.e., u + w = z. If u and w are in the same block, then u + w = w. Recently, the subject was resumed. Dress et al. [5] characterized block graphs in terms of their vertex-induced partitions: any partition of a given finite set V is a V -partition, and P V = (p v ) v∈V is a V -indexed family of V -partitions. A family P V is a compatible family of V -partitions if, for any two distinct elements u, v ∈ V , the union of the set in p v that contains u and the set in p u that contains v coincides with the set V , and {v} ∈ p v holds for all v ∈ V . Let P(V ) denote the set of all compatible families of V -partitions.
There is a bijective function between the block graphs with vertex set V and P(V ).
The graph G is a block graph if and only if there exists a unique induced path between any two vertices in G.

BACKGROUND
Basic concepts about chordal graphs are assumed to be known and can be found in Blair and Peyton [4] and Golumbic [6]. In this section, the most pertinent concepts are reviewed.
It is worth mentioning two kinds of cliques in a chordal graph G. A simplicial clique is a maximal clique containing at least one simplicial vertex. A simplicial clique Q is called a boundary clique if there exists a maximal clique Q , Q = Q , such that Q \ Q is a set of simplicial vertices of G.
It is well known that a graph G is chordal if and only if G admits a perfect elimination ordering.

BOUNDARY REPRESENTATION
In this section we present a representation of block graphs based on the concept of a perfect elimination ordering of the graph. As in a peo, where a vertex is eliminated when it is simplicial in the remaining graph, in this proposed representation, a maximal clique is eliminated when it is a boundary clique in the remaining graph. As all elements of the maximal clique are stored, the graph can be easily recovered. The representation is defined as follows; its structure is similar to the one presented in [9].
Let G = (V, E) be a block graph with maximal cliques. A boundary representation of G is the sequence of pairs BR(G) = [(P 1 , s 1 ), . . . , (P , s )] such that  The algorithm to build the representation proceeds in stages. In each stage the boundary cliques of the current graph are determined. For each boundary clique, the simplicial vertices and its corresponding cut-vertex are recorded and all the simplicial vertices are removed from the graph. The process is repeated until only one clique remains. Note that it is possible to obtain a perfect elimination ordering of G in direct sense, unlike other well known algorithms (lexicographic breadth-first search [12], for instance).
The boundary representation makes possible to deduce some structural properties of the graph. Property 3. The sequence provided by all vertices of P 1 , followed by all vertices of P 2 , and so on, up to P is a perfect elimination ordering of G. Observe that, since there is no order in the set P i , i = 1, . . . , , several sequences can be built.
Employing the algorithm for the graph in Figure 1 we have:

ONE-VERTEX EXTENSIONS
The concept of one-vertex extension was introduced by Bandelt and Mulder [1].
Let G = (V, E) be a graph, v ∈ V and u / ∈ V . An extension of G to a graph G = (V , E ) is a one-vertex extension if it obeys one of the following three rules: The special cases of (α), (β ) and (γ) restricted to a simplicial vertex v ∈ V are denoted by (α * ), (β * ) and (γ * ), respectively.
Consider a graph G, CV (G) the set of cut-vertices, Simp(G) the set of simplicial vertices and Q(G) the set of maximal cliques of the graph. Proof. Consider a block graph G with maximal cliques and its boundary representation BR(G) = [(P 1 , s 1 ), . . . , (P , )]. It is possible to construct a sequence Π(G) by transversing the boundary representation in reverse order.
has v as a pendant vertex and s − j as a cut-vertex. The vertices v and s − j belong to a new maximal clique Q. For x ∈ P − j \ {v}, let be the triple (β * , v, x). Thus, there are the following elements of the sequence Π(G): These extensions increase the clique Q to which vertex v belongs in G. Then, we obtain the one-extension sequence of G, Π(G), composed by type (α) and type (β * ) extensions. Let Π(G) = Π(H) π(n) = [π(1), . . . , π(n − 1), π(n)].
If π(n) = (α, v, u), two cases must be analyzed. In any case, G is a block graph.
Tend. Mat. Apl. Comput., 20, N. 2 (2019) The proof of Theorem 4.1 provides a possible one-extension sequence of a block graph. As an example, consider the block graph G in Figure 1

ACKNOWLEDGEMENT
This work was partially supported by CNPq grant 304706/2017-5.