1. Introduction Constrained orientations, that is orientations such that all the vertices have a prescribed indegree, relates to one another many combinatorial and topological properties such as arboricity, connectivity and planarity. These orientations are the basic tool to solve planar augmentation problems 2]. We are concerned with two classes of planar graphs: maximal planar graphs (i.e. po...

We deene the subvariance S P (F) of a family of graphs F with respect to property P to be the innmum of the ratio jH1j jH2j , where H 1 and H 2 are any two maximal spanning subgraphs of G with property P, and where G is a member of F. It is shown that, for the family of all connected graphs, the subvariance when P is planar, outerplanar, and bipartite planar, is 1=3, 1=2, and 1=2, respectively.

the paper is concerned with the bifurcation of limit cycles in general quadratic perturbations of a quadratic reversible and non-hamiltonian system, whose period annulus is bounded by an elliptic separatrix related to a singularity at infinity in the poincar'{e} disk. attention goes to the number of limit cycles produced by the period annulus under perturbations. by using the appropriate p...

A graph is hamiltonian if it has a hamiltonian cycle. It is well-known that Tutte proved that any 4connected planar graph is hamiltonian. It is also well-known that the problem of determining whether a 3-connected planar graph is hamiltonian is NP-complete. In particular, Chvátal and Wigderson had independently shown that the problem of determining whether a maximal planar graph is hamiltonian ...

A 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. A 1-planar graph on n vertices can have at most 4n− 8 edges. It is known that testing 1-planarity of a graph is NP-complete. A 1-planar embedding of a graph G is maximal, if no edge can be added without violating the 1-planarity of G. In this paper, we study combinatorial properties of maximal 1-pl...

Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of 54 -tough maximal planar graphs presented by Harant and Owens [Discrete Math. 147 (1995), 301–305]. In addition, we present two generalizations of a similar result of Tkáč who considered 1-tough maximal planar graphs [Discrete Math. 154 (1996), 321–328]; we rem...

Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. The...