30] deal with a special kind of query graph where no vertex is totally surrounded by edges, called outerplanar graph
Since each page admits an outerplanar graph
, G can be decomposed into p outerplanar graphs
It is well known that every outerplanar graph
has a vertex of degree at most 2.
Therefore, for any outerplanar graph
there exists an ordering of its vertices in which the edges do not cross.
k] be the limit probability that a vertex of a connected outerplanar graph
has degree k then
Let H be a maximal outerplanar graph
(no edge can be added without violating outerplanarity); then it is immediate that the boundary of the exterior face of H is a Hamiltonian cycle C and each interior face is triangular (Harary, 1972, p.
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].
However, Astratian and Oksimets  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.
In this particular case, we provide constructive representations for interval, block and outerplanar graphs
This paradigm was used successfully in calculating the genus distributions of 3-regular outerplanar graphs
[Gr11b], of 4-regular outerplanar graphs
[PKG11], of Halin graphs [Gr13], and of the 3 x n-mesh graphs [KPG12].
Part (a) and Part (b) have been proved for outerplanar graphs
, and graphs with [DELTA] [greater than or equal to] 12 which can be embedded in a surface of nonnegative characteristic .