Extremal Parameters in Sub-Critical Graph Classes

Abstract

We analyze several extremal parameters like the diameter or the maximum degree in sub-critial graph classes. Sub-critical graph classes cover several well-known classes of graphs like trees, outerplanar graph or series-parallel graphs which have been intensively studied during the last few years. However, this paper is the first one, where these kind of parameters are studied from a general point of view.

Maximum degree in minor-closed classes of graphs

“…Examples show that this is not true for arbitrary H (see the discussion below). Using analytic methods, this upper bound can be proved for so-called subcritical classes of graphs (see [6]), which include outerplanar and series-parallel graphs.…”

Spanning trees in random series-parallel graphs

“…Finally, we analyse the generating function C(x, y) of connected SP graphs carrying a distinguished spanning tree. Since the singular expansion of B(x, y) is of a square-root type with exponent 1/2 as it is shown in Equation (19), we get the singular expansion of C(x, y) immediately from Proposition 3.10 in [25] (see also [21]). Additionally, when y = 1 we have ρ(1) ≈ 0.05288, C 0 (1) ≈ 0.05450, C 2 (1) = −τ ≈ −0.05668, and C 3 (1) ≈ 0.00145.…”

“…The precise computational method to obtain the exponential growth constant of the expected value of X n,µ as a function of the edge density is the following. For a given density µ we use (21) to obtain the corresponding y 0 . Then we use the implicit expression of the singularity curve stated in Theorem 2.2. of [8] in order to obtain the growth constant of the number of 2-connected SP graphs of edge density equal to µ.…”

“…Indeed, the family of SP graphs is the prototype of the so-called subcritical graph class family (see e.g. [21,25]). In a typical connected graph in such a family, maximal 2-connected subgraphs (also called blocks) are small compared with the total size of the graph.…”

“…The maximum degree and the degree sequence of a random SP graph have been studied in [20] and [6,19], respectively. Drmota and Noy [21] investigated several extremal parameters in subcritical graph classes, which include the class of SP graphs. They showed, for instance, that the expected diameter D n of a random connected SP graph on n vertices satisfies c 1 √ n ≤ E[D n ] ≤ c 2 √ n log n for some positive integers c 1 and c 2 .…”