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We describe how the 5-regular simple planar graphs can all be obtained from an elementary family of starting graphs by repeatedly applying a few local expansion operations. The proof uses an amalgam of theory and computation. By incorporating the recursion into the canonical construction path method of isomorph rejection, a generator of non-isomorphic embedded 5-regular planar graphs is obtained with time complexity O(n 2 ) per isomorphism class. A similar result is obtained for simple planar pentangulations with minimum degree 2.

Rectangular drawings and rectangular duals can be naturally extended to other surfaces. In this paper, we extend rectangular drawings and rectangular duals to drawings on a cylinder. The extended drawings are called rectangular-radial drawings and rectangular-radial duals. Rectangular-radial drawings correspond to periodic rectangular tilings of a 1-dimensional strip. We establish a necessary and sufficient condition for plane graphs with maximum degree 3 to have rectangular-radial drawings and a necessary and sufficient condition for triangulated plane graphs to have rectangular-radial duals. Furthermore, we present three linear time algorithms under three different conditions for finding a rectangular-radial drawing for a given cubic plane graph, if one exists.

Let G = (V, E) be a simple and undirected graph. For some integer k 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardinality of a minimum k-dominating set and a minimum α-dominating set in G is said to be the k-domination number and the α-domination number of G, respectively. In this paper, we present some approximability and inapproximability results on the problem of finding of k-domination number and α-domination number of some classes of graphs. Moreover, we introduce a generalization of α-dominating set which we call f -dominating set. Given a function f : N → R, where N = {1, 2, 3, . . .}, a set D ⊆ V is said to be an f -dominating set in G if every vertex v of G outside D has at least f (dv) neighbors in D. We prove NP-hardness of the problem of finding of a minimum f -dominating set in G, for a large family of functions f .

Abstract. The method of switchings is a standard tool for enumerative and probabilistic applications in combinatorics. In its simplest form, it analyses a relation between two sets to estimate the ratio of their sizes. In a more complicated setting, there is a family of sets connected by some relations. By bounding properties of the relations, bounds can be inferred on the relative sizes of the sets. In this paper we extend the treatment of Fack and McKay (2007) to allow the graph of sets and relations to be an arbitrary directed graph. A special case that frequently occurs in bounding tails of distributions is analysed in detail.