A graph is 1-planar if it can be drawn in the plane such that each of its edges is crossed at most once. We prove a conjecture of Czap and Hudák [6] stating that the edge set of every 1-planar graph can be decomposed into a planar graph and a forest. We also provide simple proofs for the following recent results: (i) an n-vertex graph that admits a 1-planar drawing with straight-line edges has ...

In this paper, the composite order Cayley graph Cay(G, S) is introduced, where G is a group and S is the set of all composite order elements of G. Some graph parameters such as diameter, girth, clique number, independence number, vertex chromatic number and domination number are calculated for the composite order Cayley graph of some certain groups. Moreover, the planarity of composite order Ca...

A graph is k-planar (k ≥ 1) if it can be drawn in the plane such that no edge is crossed more than k times. A graph is k-quasi planar (k ≥ 2) if it can be drawn in the plane with no k pairwise crossing edges. The families of k-planar and k-quasi planar graphs have been widely studied in the literature, and several bounds have been proven on their edge density. Nonetheless, only trivial results ...

We show that every n-vertex planar graph admits a simultaneous embedding with no mapping and with fixed edges with any (n/2)-vertex planar graph. In order to achieve this result, we prove that every n-vertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every n-vertex planar graph and every n-vertex planar partial 3-tree admit a simultaneous...

This paper is dedicated to propose an algorithm in order to generate the certain isomers of some well-known fullerene bases. Furthermore, we enlist the bipartite edge frustration correlated with some of symmetrically distinct innite families of fullerenes generated by the oered process.

We show that every n-vertex planar graph admits a simultaneous embedding without mapping and with fixed edges with any (n/2)vertex planar graph. In order to achieve this result, we prove that every n-vertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every n-vertex planar graph and every n-vertex planar partial 3-tree admit a simultaneous ...

This paper presented study on convex drawing of planar graph. In graph theory, a planar graph is a graph that can be embedded in the plane. A planar graph is one that can be drawn on a plane in such a way that there are no “edge crossings,” i.e. edges intersect only at their common vertices. Convex polygon has all interior angles less than or equal to 180°. A graph is called a convex drawing if...