In second place, Yamasaki et al., [30] deal with a special kind of query graph where no vertex is totally surrounded by edges, called

outerplanar graph pattern.

The next lemma makes use of the following property of

outerplanar graphs. If G = (V,E) is an

outerplanar graph, then |E|[less than or equal to]2|V|-3.

Moreover, G is an

outerplanar graph. It is well known that every

outerplanar graph has a vertex of degree at most 2.

Proof: Let [member of] be an

outerplanar graph. We shall show that for every maximal biconnected component H of [member of] and a vertex u in it, there exists a c-p-BOX(1)-realization of H in which u is safe.

Proof: Let G be an

outerplanar graph. We shall show that for every maximal biconnected component H of G and a vertex u in it, there exists a c-p-BOX(1)-realization of H in which u is safe.

Theorem 3.2 Let [d.sub.k] be the limit probability that a vertex of a connected

outerplanar graph has degree k then

A dissection is a biconnected

outerplanar graph. All these graphs are well-known and well-studied combinatorial objects [FS05, DFLS04, FN99], which have plenty of applications in physics, computer science, and bioinformatics (see e.g.

The "dual" construction is also possible: given any undirected tree whose all internal vertices have degree three, one can obtain a maximal

outerplanar graph by embedding the tree in the plane, collapsing all the leaves into a single vertex, obtaining a plane multigraph, and taking the planar dual of this multigraph (Bondy and Murty, 1976, Sec.

Theorem 1.11 For all i [greater than or equal to] 4, there exists an

outerplanar graph G without cycles of lengths 4 to i such that [[chi].sub.0](G) [greater than or equal to] 7.

However, Astratian and Oksimets [2] showed that LT graphs that are not maximal outerplanar (maximal

outerplanar graphs are hamiltonian) have size at least 2n - 2.

Kierstead, "The relaxed game chromatic number of

outerplanar graphs," Journal of Graph Theory, 46 (2004) 69-78.

Wang, "Adjacent vertex distinguishing total colorings of

outerplanar graphs," Journal of Combinatorial Optimization, vol.